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Theorem alexsubALTlem2 23415
Description: Lemma for alexsubALT 23418. Every subset of a base which has no finite subcover is a subset of a maximal such collection. (Contributed by Jeff Hankins, 27-Jan-2010.)
Hypothesis
Ref Expression
alexsubALT.1 𝑋 = βˆͺ 𝐽
Assertion
Ref Expression
alexsubALTlem2 (((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) β†’ βˆƒπ‘’ ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…})βˆ€π‘£ ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) Β¬ 𝑒 ⊊ 𝑣)
Distinct variable groups:   π‘Ž,𝑏,𝑐,𝑑,𝑒,𝑣,π‘₯,𝑧,𝐽   𝑋,π‘Ž,𝑏,𝑐,𝑑,𝑒,𝑣,π‘₯,𝑧

Proof of Theorem alexsubALTlem2
Dummy variables 𝑛 𝑀 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssel 3942 . . . . . . . . . . . . 13 (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) β†’ (𝑀 ∈ 𝑦 β†’ 𝑀 ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…})))
2 elun 4113 . . . . . . . . . . . . . . 15 (𝑀 ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ↔ (𝑀 ∈ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} ∨ 𝑀 ∈ {βˆ…}))
3 sseq2 3975 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑀 β†’ (π‘Ž βŠ† 𝑧 ↔ π‘Ž βŠ† 𝑀))
4 pweq 4579 . . . . . . . . . . . . . . . . . . . 20 (𝑧 = 𝑀 β†’ 𝒫 𝑧 = 𝒫 𝑀)
54ineq1d 4176 . . . . . . . . . . . . . . . . . . 19 (𝑧 = 𝑀 β†’ (𝒫 𝑧 ∩ Fin) = (𝒫 𝑀 ∩ Fin))
65raleqdv 3316 . . . . . . . . . . . . . . . . . 18 (𝑧 = 𝑀 β†’ (βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏 ↔ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏))
73, 6anbi12d 632 . . . . . . . . . . . . . . . . 17 (𝑧 = 𝑀 β†’ ((π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ↔ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)))
87elrab 3650 . . . . . . . . . . . . . . . 16 (𝑀 ∈ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} ↔ (𝑀 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)))
9 velsn 4607 . . . . . . . . . . . . . . . 16 (𝑀 ∈ {βˆ…} ↔ 𝑀 = βˆ…)
108, 9orbi12i 914 . . . . . . . . . . . . . . 15 ((𝑀 ∈ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} ∨ 𝑀 ∈ {βˆ…}) ↔ ((𝑀 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)) ∨ 𝑀 = βˆ…))
112, 10bitri 275 . . . . . . . . . . . . . 14 (𝑀 ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ↔ ((𝑀 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)) ∨ 𝑀 = βˆ…))
12 elpwi 4572 . . . . . . . . . . . . . . . 16 (𝑀 ∈ 𝒫 (fiβ€˜π‘₯) β†’ 𝑀 βŠ† (fiβ€˜π‘₯))
1312adantr 482 . . . . . . . . . . . . . . 15 ((𝑀 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)) β†’ 𝑀 βŠ† (fiβ€˜π‘₯))
14 0ss 4361 . . . . . . . . . . . . . . . 16 βˆ… βŠ† (fiβ€˜π‘₯)
15 sseq1 3974 . . . . . . . . . . . . . . . 16 (𝑀 = βˆ… β†’ (𝑀 βŠ† (fiβ€˜π‘₯) ↔ βˆ… βŠ† (fiβ€˜π‘₯)))
1614, 15mpbiri 258 . . . . . . . . . . . . . . 15 (𝑀 = βˆ… β†’ 𝑀 βŠ† (fiβ€˜π‘₯))
1713, 16jaoi 856 . . . . . . . . . . . . . 14 (((𝑀 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)) ∨ 𝑀 = βˆ…) β†’ 𝑀 βŠ† (fiβ€˜π‘₯))
1811, 17sylbi 216 . . . . . . . . . . . . 13 (𝑀 ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) β†’ 𝑀 βŠ† (fiβ€˜π‘₯))
191, 18syl6 35 . . . . . . . . . . . 12 (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) β†’ (𝑀 ∈ 𝑦 β†’ 𝑀 βŠ† (fiβ€˜π‘₯)))
2019ralrimiv 3143 . . . . . . . . . . 11 (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) β†’ βˆ€π‘€ ∈ 𝑦 𝑀 βŠ† (fiβ€˜π‘₯))
21 unissb 4905 . . . . . . . . . . 11 (βˆͺ 𝑦 βŠ† (fiβ€˜π‘₯) ↔ βˆ€π‘€ ∈ 𝑦 𝑀 βŠ† (fiβ€˜π‘₯))
2220, 21sylibr 233 . . . . . . . . . 10 (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) β†’ βˆͺ 𝑦 βŠ† (fiβ€˜π‘₯))
2322adantr 482 . . . . . . . . 9 ((𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦) β†’ βˆͺ 𝑦 βŠ† (fiβ€˜π‘₯))
2423ad2antlr 726 . . . . . . . 8 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ Β¬ βˆͺ 𝑦 = βˆ…) β†’ βˆͺ 𝑦 βŠ† (fiβ€˜π‘₯))
25 vuniex 7681 . . . . . . . . 9 βˆͺ 𝑦 ∈ V
2625elpw 4569 . . . . . . . 8 (βˆͺ 𝑦 ∈ 𝒫 (fiβ€˜π‘₯) ↔ βˆͺ 𝑦 βŠ† (fiβ€˜π‘₯))
2724, 26sylibr 233 . . . . . . 7 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ Β¬ βˆͺ 𝑦 = βˆ…) β†’ βˆͺ 𝑦 ∈ 𝒫 (fiβ€˜π‘₯))
28 uni0b 4899 . . . . . . . . . 10 (βˆͺ 𝑦 = βˆ… ↔ 𝑦 βŠ† {βˆ…})
2928notbii 320 . . . . . . . . 9 (Β¬ βˆͺ 𝑦 = βˆ… ↔ Β¬ 𝑦 βŠ† {βˆ…})
30 disjssun 4432 . . . . . . . . . . . . 13 ((𝑦 ∩ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}) = βˆ… β†’ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ↔ 𝑦 βŠ† {βˆ…}))
3130biimpcd 249 . . . . . . . . . . . 12 (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) β†’ ((𝑦 ∩ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}) = βˆ… β†’ 𝑦 βŠ† {βˆ…}))
3231necon3bd 2958 . . . . . . . . . . 11 (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) β†’ (Β¬ 𝑦 βŠ† {βˆ…} β†’ (𝑦 ∩ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}) β‰  βˆ…))
33 n0 4311 . . . . . . . . . . . 12 ((𝑦 ∩ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}) β‰  βˆ… ↔ βˆƒπ‘€ 𝑀 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}))
34 elin 3931 . . . . . . . . . . . . . . 15 (𝑀 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}) ↔ (𝑀 ∈ 𝑦 ∧ 𝑀 ∈ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}))
358anbi2i 624 . . . . . . . . . . . . . . 15 ((𝑀 ∈ 𝑦 ∧ 𝑀 ∈ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}) ↔ (𝑀 ∈ 𝑦 ∧ (𝑀 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏))))
3634, 35bitri 275 . . . . . . . . . . . . . 14 (𝑀 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}) ↔ (𝑀 ∈ 𝑦 ∧ (𝑀 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏))))
37 simprrl 780 . . . . . . . . . . . . . . 15 ((𝑀 ∈ 𝑦 ∧ (𝑀 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏))) β†’ π‘Ž βŠ† 𝑀)
38 simpl 484 . . . . . . . . . . . . . . 15 ((𝑀 ∈ 𝑦 ∧ (𝑀 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏))) β†’ 𝑀 ∈ 𝑦)
39 ssuni 4898 . . . . . . . . . . . . . . 15 ((π‘Ž βŠ† 𝑀 ∧ 𝑀 ∈ 𝑦) β†’ π‘Ž βŠ† βˆͺ 𝑦)
4037, 38, 39syl2anc 585 . . . . . . . . . . . . . 14 ((𝑀 ∈ 𝑦 ∧ (𝑀 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏))) β†’ π‘Ž βŠ† βˆͺ 𝑦)
4136, 40sylbi 216 . . . . . . . . . . . . 13 (𝑀 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}) β†’ π‘Ž βŠ† βˆͺ 𝑦)
4241exlimiv 1934 . . . . . . . . . . . 12 (βˆƒπ‘€ 𝑀 ∈ (𝑦 ∩ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}) β†’ π‘Ž βŠ† βˆͺ 𝑦)
4333, 42sylbi 216 . . . . . . . . . . 11 ((𝑦 ∩ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}) β‰  βˆ… β†’ π‘Ž βŠ† βˆͺ 𝑦)
4432, 43syl6 35 . . . . . . . . . 10 (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) β†’ (Β¬ 𝑦 βŠ† {βˆ…} β†’ π‘Ž βŠ† βˆͺ 𝑦))
4544ad2antrl 727 . . . . . . . . 9 ((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) β†’ (Β¬ 𝑦 βŠ† {βˆ…} β†’ π‘Ž βŠ† βˆͺ 𝑦))
4629, 45biimtrid 241 . . . . . . . 8 ((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) β†’ (Β¬ βˆͺ 𝑦 = βˆ… β†’ π‘Ž βŠ† βˆͺ 𝑦))
4746imp 408 . . . . . . 7 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ Β¬ βˆͺ 𝑦 = βˆ…) β†’ π‘Ž βŠ† βˆͺ 𝑦)
48 elfpw 9305 . . . . . . . . . 10 (𝑛 ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) ↔ (𝑛 βŠ† βˆͺ 𝑦 ∧ 𝑛 ∈ Fin))
49 unieq 4881 . . . . . . . . . . . . . . . . . . 19 (𝑦 = βˆ… β†’ βˆͺ 𝑦 = βˆͺ βˆ…)
50 uni0 4901 . . . . . . . . . . . . . . . . . . 19 βˆͺ βˆ… = βˆ…
5149, 50eqtrdi 2793 . . . . . . . . . . . . . . . . . 18 (𝑦 = βˆ… β†’ βˆͺ 𝑦 = βˆ…)
5251necon3bi 2971 . . . . . . . . . . . . . . . . 17 (Β¬ βˆͺ 𝑦 = βˆ… β†’ 𝑦 β‰  βˆ…)
5352adantr 482 . . . . . . . . . . . . . . . 16 ((Β¬ βˆͺ 𝑦 = βˆ… ∧ 𝑛 ∈ Fin) β†’ 𝑦 β‰  βˆ…)
5453ad2antrl 727 . . . . . . . . . . . . . . 15 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ ((Β¬ βˆͺ 𝑦 = βˆ… ∧ 𝑛 ∈ Fin) ∧ 𝑛 βŠ† βˆͺ 𝑦)) β†’ 𝑦 β‰  βˆ…)
55 simplrr 777 . . . . . . . . . . . . . . 15 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ ((Β¬ βˆͺ 𝑦 = βˆ… ∧ 𝑛 ∈ Fin) ∧ 𝑛 βŠ† βˆͺ 𝑦)) β†’ [⊊] Or 𝑦)
56 simprlr 779 . . . . . . . . . . . . . . 15 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ ((Β¬ βˆͺ 𝑦 = βˆ… ∧ 𝑛 ∈ Fin) ∧ 𝑛 βŠ† βˆͺ 𝑦)) β†’ 𝑛 ∈ Fin)
57 simprr 772 . . . . . . . . . . . . . . 15 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ ((Β¬ βˆͺ 𝑦 = βˆ… ∧ 𝑛 ∈ Fin) ∧ 𝑛 βŠ† βˆͺ 𝑦)) β†’ 𝑛 βŠ† βˆͺ 𝑦)
58 finsschain 9310 . . . . . . . . . . . . . . 15 (((𝑦 β‰  βˆ… ∧ [⊊] Or 𝑦) ∧ (𝑛 ∈ Fin ∧ 𝑛 βŠ† βˆͺ 𝑦)) β†’ βˆƒπ‘€ ∈ 𝑦 𝑛 βŠ† 𝑀)
5954, 55, 56, 57, 58syl22anc 838 . . . . . . . . . . . . . 14 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ ((Β¬ βˆͺ 𝑦 = βˆ… ∧ 𝑛 ∈ Fin) ∧ 𝑛 βŠ† βˆͺ 𝑦)) β†’ βˆƒπ‘€ ∈ 𝑦 𝑛 βŠ† 𝑀)
6059expr 458 . . . . . . . . . . . . 13 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ (Β¬ βˆͺ 𝑦 = βˆ… ∧ 𝑛 ∈ Fin)) β†’ (𝑛 βŠ† βˆͺ 𝑦 β†’ βˆƒπ‘€ ∈ 𝑦 𝑛 βŠ† 𝑀))
61 0elpw 5316 . . . . . . . . . . . . . . . . . . . 20 βˆ… ∈ 𝒫 π‘Ž
62 0fin 9122 . . . . . . . . . . . . . . . . . . . 20 βˆ… ∈ Fin
6361, 62elini 4158 . . . . . . . . . . . . . . . . . . 19 βˆ… ∈ (𝒫 π‘Ž ∩ Fin)
64 unieq 4881 . . . . . . . . . . . . . . . . . . . . . 22 (𝑏 = βˆ… β†’ βˆͺ 𝑏 = βˆͺ βˆ…)
6564eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . 21 (𝑏 = βˆ… β†’ (𝑋 = βˆͺ 𝑏 ↔ 𝑋 = βˆͺ βˆ…))
6665notbid 318 . . . . . . . . . . . . . . . . . . . 20 (𝑏 = βˆ… β†’ (Β¬ 𝑋 = βˆͺ 𝑏 ↔ Β¬ 𝑋 = βˆͺ βˆ…))
6766rspccv 3581 . . . . . . . . . . . . . . . . . . 19 (βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏 β†’ (βˆ… ∈ (𝒫 π‘Ž ∩ Fin) β†’ Β¬ 𝑋 = βˆͺ βˆ…))
6863, 67mpi 20 . . . . . . . . . . . . . . . . . 18 (βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏 β†’ Β¬ 𝑋 = βˆͺ βˆ…)
69 velpw 4570 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 ∈ 𝒫 𝑀 ↔ 𝑛 βŠ† 𝑀)
70 elin 3931 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 ∈ (𝒫 𝑀 ∩ Fin) ↔ (𝑛 ∈ 𝒫 𝑀 ∧ 𝑛 ∈ Fin))
71 unieq 4881 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑏 = 𝑛 β†’ βˆͺ 𝑏 = βˆͺ 𝑛)
7271eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (𝑏 = 𝑛 β†’ (𝑋 = βˆͺ 𝑏 ↔ 𝑋 = βˆͺ 𝑛))
7372notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (𝑏 = 𝑛 β†’ (Β¬ 𝑋 = βˆͺ 𝑏 ↔ Β¬ 𝑋 = βˆͺ 𝑛))
7473rspccv 3581 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏 β†’ (𝑛 ∈ (𝒫 𝑀 ∩ Fin) β†’ Β¬ 𝑋 = βˆͺ 𝑛))
7570, 74biimtrrid 242 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏 β†’ ((𝑛 ∈ 𝒫 𝑀 ∧ 𝑛 ∈ Fin) β†’ Β¬ 𝑋 = βˆͺ 𝑛))
7675expd 417 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏 β†’ (𝑛 ∈ 𝒫 𝑀 β†’ (𝑛 ∈ Fin β†’ Β¬ 𝑋 = βˆͺ 𝑛)))
7769, 76biimtrrid 242 . . . . . . . . . . . . . . . . . . . . . . . . 25 (βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏 β†’ (𝑛 βŠ† 𝑀 β†’ (𝑛 ∈ Fin β†’ Β¬ 𝑋 = βˆͺ 𝑛)))
7877com23 86 . . . . . . . . . . . . . . . . . . . . . . . 24 (βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏 β†’ (𝑛 ∈ Fin β†’ (𝑛 βŠ† 𝑀 β†’ Β¬ 𝑋 = βˆͺ 𝑛)))
7978ad2antll 728 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑀 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)) β†’ (𝑛 ∈ Fin β†’ (𝑛 βŠ† 𝑀 β†’ Β¬ 𝑋 = βˆͺ 𝑛)))
8079a1i 11 . . . . . . . . . . . . . . . . . . . . . 22 (Β¬ 𝑋 = βˆͺ βˆ… β†’ ((𝑀 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)) β†’ (𝑛 ∈ Fin β†’ (𝑛 βŠ† 𝑀 β†’ Β¬ 𝑋 = βˆͺ 𝑛))))
81 sseq2 3975 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑀 = βˆ… β†’ (𝑛 βŠ† 𝑀 ↔ 𝑛 βŠ† βˆ…))
82 ss0 4363 . . . . . . . . . . . . . . . . . . . . . . . . 25 (𝑛 βŠ† βˆ… β†’ 𝑛 = βˆ…)
8381, 82syl6bi 253 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑀 = βˆ… β†’ (𝑛 βŠ† 𝑀 β†’ 𝑛 = βˆ…))
84 unieq 4881 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑛 = βˆ… β†’ βˆͺ 𝑛 = βˆͺ βˆ…)
8584eqeq2d 2748 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝑛 = βˆ… β†’ (𝑋 = βˆͺ 𝑛 ↔ 𝑋 = βˆͺ βˆ…))
8685notbid 318 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑛 = βˆ… β†’ (Β¬ 𝑋 = βˆͺ 𝑛 ↔ Β¬ 𝑋 = βˆͺ βˆ…))
8786biimprcd 250 . . . . . . . . . . . . . . . . . . . . . . . . 25 (Β¬ 𝑋 = βˆͺ βˆ… β†’ (𝑛 = βˆ… β†’ Β¬ 𝑋 = βˆͺ 𝑛))
8887a1dd 50 . . . . . . . . . . . . . . . . . . . . . . . 24 (Β¬ 𝑋 = βˆͺ βˆ… β†’ (𝑛 = βˆ… β†’ (𝑛 ∈ Fin β†’ Β¬ 𝑋 = βˆͺ 𝑛)))
8983, 88syl9r 78 . . . . . . . . . . . . . . . . . . . . . . 23 (Β¬ 𝑋 = βˆͺ βˆ… β†’ (𝑀 = βˆ… β†’ (𝑛 βŠ† 𝑀 β†’ (𝑛 ∈ Fin β†’ Β¬ 𝑋 = βˆͺ 𝑛))))
9089com34 91 . . . . . . . . . . . . . . . . . . . . . 22 (Β¬ 𝑋 = βˆͺ βˆ… β†’ (𝑀 = βˆ… β†’ (𝑛 ∈ Fin β†’ (𝑛 βŠ† 𝑀 β†’ Β¬ 𝑋 = βˆͺ 𝑛))))
9180, 90jaod 858 . . . . . . . . . . . . . . . . . . . . 21 (Β¬ 𝑋 = βˆͺ βˆ… β†’ (((𝑀 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† 𝑀 ∧ βˆ€π‘ ∈ (𝒫 𝑀 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)) ∨ 𝑀 = βˆ…) β†’ (𝑛 ∈ Fin β†’ (𝑛 βŠ† 𝑀 β†’ Β¬ 𝑋 = βˆͺ 𝑛))))
9211, 91biimtrid 241 . . . . . . . . . . . . . . . . . . . 20 (Β¬ 𝑋 = βˆͺ βˆ… β†’ (𝑀 ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) β†’ (𝑛 ∈ Fin β†’ (𝑛 βŠ† 𝑀 β†’ Β¬ 𝑋 = βˆͺ 𝑛))))
931, 92sylan9r 510 . . . . . . . . . . . . . . . . . . 19 ((Β¬ 𝑋 = βˆͺ βˆ… ∧ 𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…})) β†’ (𝑀 ∈ 𝑦 β†’ (𝑛 ∈ Fin β†’ (𝑛 βŠ† 𝑀 β†’ Β¬ 𝑋 = βˆͺ 𝑛))))
9493com23 86 . . . . . . . . . . . . . . . . . 18 ((Β¬ 𝑋 = βˆͺ βˆ… ∧ 𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…})) β†’ (𝑛 ∈ Fin β†’ (𝑀 ∈ 𝑦 β†’ (𝑛 βŠ† 𝑀 β†’ Β¬ 𝑋 = βˆͺ 𝑛))))
9568, 94sylan 581 . . . . . . . . . . . . . . . . 17 ((βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏 ∧ 𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…})) β†’ (𝑛 ∈ Fin β†’ (𝑀 ∈ 𝑦 β†’ (𝑛 βŠ† 𝑀 β†’ Β¬ 𝑋 = βˆͺ 𝑛))))
9695ad2ant2lr 747 . . . . . . . . . . . . . . . 16 ((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) β†’ (𝑛 ∈ Fin β†’ (𝑀 ∈ 𝑦 β†’ (𝑛 βŠ† 𝑀 β†’ Β¬ 𝑋 = βˆͺ 𝑛))))
9796imp 408 . . . . . . . . . . . . . . 15 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ 𝑛 ∈ Fin) β†’ (𝑀 ∈ 𝑦 β†’ (𝑛 βŠ† 𝑀 β†’ Β¬ 𝑋 = βˆͺ 𝑛)))
9897adantrl 715 . . . . . . . . . . . . . 14 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ (Β¬ βˆͺ 𝑦 = βˆ… ∧ 𝑛 ∈ Fin)) β†’ (𝑀 ∈ 𝑦 β†’ (𝑛 βŠ† 𝑀 β†’ Β¬ 𝑋 = βˆͺ 𝑛)))
9998rexlimdv 3151 . . . . . . . . . . . . 13 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ (Β¬ βˆͺ 𝑦 = βˆ… ∧ 𝑛 ∈ Fin)) β†’ (βˆƒπ‘€ ∈ 𝑦 𝑛 βŠ† 𝑀 β†’ Β¬ 𝑋 = βˆͺ 𝑛))
10060, 99syld 47 . . . . . . . . . . . 12 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ (Β¬ βˆͺ 𝑦 = βˆ… ∧ 𝑛 ∈ Fin)) β†’ (𝑛 βŠ† βˆͺ 𝑦 β†’ Β¬ 𝑋 = βˆͺ 𝑛))
101100expr 458 . . . . . . . . . . 11 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ Β¬ βˆͺ 𝑦 = βˆ…) β†’ (𝑛 ∈ Fin β†’ (𝑛 βŠ† βˆͺ 𝑦 β†’ Β¬ 𝑋 = βˆͺ 𝑛)))
102101impcomd 413 . . . . . . . . . 10 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ Β¬ βˆͺ 𝑦 = βˆ…) β†’ ((𝑛 βŠ† βˆͺ 𝑦 ∧ 𝑛 ∈ Fin) β†’ Β¬ 𝑋 = βˆͺ 𝑛))
10348, 102biimtrid 241 . . . . . . . . 9 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ Β¬ βˆͺ 𝑦 = βˆ…) β†’ (𝑛 ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) β†’ Β¬ 𝑋 = βˆͺ 𝑛))
104103ralrimiv 3143 . . . . . . . 8 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ Β¬ βˆͺ 𝑦 = βˆ…) β†’ βˆ€π‘› ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑛)
105 unieq 4881 . . . . . . . . . . 11 (𝑛 = 𝑏 β†’ βˆͺ 𝑛 = βˆͺ 𝑏)
106105eqeq2d 2748 . . . . . . . . . 10 (𝑛 = 𝑏 β†’ (𝑋 = βˆͺ 𝑛 ↔ 𝑋 = βˆͺ 𝑏))
107106notbid 318 . . . . . . . . 9 (𝑛 = 𝑏 β†’ (Β¬ 𝑋 = βˆͺ 𝑛 ↔ Β¬ 𝑋 = βˆͺ 𝑏))
108107cbvralvw 3228 . . . . . . . 8 (βˆ€π‘› ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑛 ↔ βˆ€π‘ ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)
109104, 108sylib 217 . . . . . . 7 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ Β¬ βˆͺ 𝑦 = βˆ…) β†’ βˆ€π‘ ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)
11027, 47, 109jca32 517 . . . . . 6 (((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) ∧ Β¬ βˆͺ 𝑦 = βˆ…) β†’ (βˆͺ 𝑦 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† βˆͺ 𝑦 ∧ βˆ€π‘ ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)))
111110ex 414 . . . . 5 ((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) β†’ (Β¬ βˆͺ 𝑦 = βˆ… β†’ (βˆͺ 𝑦 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† βˆͺ 𝑦 ∧ βˆ€π‘ ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏))))
112 orcom 869 . . . . . 6 ((βˆͺ 𝑦 ∈ {βˆ…} ∨ βˆͺ 𝑦 ∈ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}) ↔ (βˆͺ 𝑦 ∈ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} ∨ βˆͺ 𝑦 ∈ {βˆ…}))
11325elsn 4606 . . . . . . . 8 (βˆͺ 𝑦 ∈ {βˆ…} ↔ βˆͺ 𝑦 = βˆ…)
114 sseq2 3975 . . . . . . . . . 10 (𝑧 = βˆͺ 𝑦 β†’ (π‘Ž βŠ† 𝑧 ↔ π‘Ž βŠ† βˆͺ 𝑦))
115 pweq 4579 . . . . . . . . . . . 12 (𝑧 = βˆͺ 𝑦 β†’ 𝒫 𝑧 = 𝒫 βˆͺ 𝑦)
116115ineq1d 4176 . . . . . . . . . . 11 (𝑧 = βˆͺ 𝑦 β†’ (𝒫 𝑧 ∩ Fin) = (𝒫 βˆͺ 𝑦 ∩ Fin))
117116raleqdv 3316 . . . . . . . . . 10 (𝑧 = βˆͺ 𝑦 β†’ (βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏 ↔ βˆ€π‘ ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏))
118114, 117anbi12d 632 . . . . . . . . 9 (𝑧 = βˆͺ 𝑦 β†’ ((π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ↔ (π‘Ž βŠ† βˆͺ 𝑦 ∧ βˆ€π‘ ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)))
119118elrab 3650 . . . . . . . 8 (βˆͺ 𝑦 ∈ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} ↔ (βˆͺ 𝑦 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† βˆͺ 𝑦 ∧ βˆ€π‘ ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)))
120113, 119orbi12i 914 . . . . . . 7 ((βˆͺ 𝑦 ∈ {βˆ…} ∨ βˆͺ 𝑦 ∈ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}) ↔ (βˆͺ 𝑦 = βˆ… ∨ (βˆͺ 𝑦 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† βˆͺ 𝑦 ∧ βˆ€π‘ ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏))))
121 df-or 847 . . . . . . 7 ((βˆͺ 𝑦 = βˆ… ∨ (βˆͺ 𝑦 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† βˆͺ 𝑦 ∧ βˆ€π‘ ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏))) ↔ (Β¬ βˆͺ 𝑦 = βˆ… β†’ (βˆͺ 𝑦 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† βˆͺ 𝑦 ∧ βˆ€π‘ ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏))))
122120, 121bitr2i 276 . . . . . 6 ((Β¬ βˆͺ 𝑦 = βˆ… β†’ (βˆͺ 𝑦 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† βˆͺ 𝑦 ∧ βˆ€π‘ ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏))) ↔ (βˆͺ 𝑦 ∈ {βˆ…} ∨ βˆͺ 𝑦 ∈ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)}))
123 elun 4113 . . . . . 6 (βˆͺ 𝑦 ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ↔ (βˆͺ 𝑦 ∈ {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} ∨ βˆͺ 𝑦 ∈ {βˆ…}))
124112, 122, 1233bitr4i 303 . . . . 5 ((Β¬ βˆͺ 𝑦 = βˆ… β†’ (βˆͺ 𝑦 ∈ 𝒫 (fiβ€˜π‘₯) ∧ (π‘Ž βŠ† βˆͺ 𝑦 ∧ βˆ€π‘ ∈ (𝒫 βˆͺ 𝑦 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏))) ↔ βˆͺ 𝑦 ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}))
125111, 124sylib 217 . . . 4 ((((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) ∧ (𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦)) β†’ βˆͺ 𝑦 ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}))
126125ex 414 . . 3 (((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) β†’ ((𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦) β†’ βˆͺ 𝑦 ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…})))
127126alrimiv 1931 . 2 (((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) β†’ βˆ€π‘¦((𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦) β†’ βˆͺ 𝑦 ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…})))
128 fvex 6860 . . . . . 6 (fiβ€˜π‘₯) ∈ V
129128pwex 5340 . . . . 5 𝒫 (fiβ€˜π‘₯) ∈ V
130129rabex 5294 . . . 4 {𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} ∈ V
131 p0ex 5344 . . . 4 {βˆ…} ∈ V
132130, 131unex 7685 . . 3 ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∈ V
133132zorn 10450 . 2 (βˆ€π‘¦((𝑦 βŠ† ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) ∧ [⊊] Or 𝑦) β†’ βˆͺ 𝑦 ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…})) β†’ βˆƒπ‘’ ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…})βˆ€π‘£ ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) Β¬ 𝑒 ⊊ 𝑣)
134127, 133syl 17 1 (((𝐽 = (topGenβ€˜(fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ 𝒫 π‘₯(𝑋 = βˆͺ 𝑐 β†’ βˆƒπ‘‘ ∈ (𝒫 𝑐 ∩ Fin)𝑋 = βˆͺ 𝑑) ∧ π‘Ž ∈ 𝒫 (fiβ€˜π‘₯)) ∧ βˆ€π‘ ∈ (𝒫 π‘Ž ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏) β†’ βˆƒπ‘’ ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…})βˆ€π‘£ ∈ ({𝑧 ∈ 𝒫 (fiβ€˜π‘₯) ∣ (π‘Ž βŠ† 𝑧 ∧ βˆ€π‘ ∈ (𝒫 𝑧 ∩ Fin) Β¬ 𝑋 = βˆͺ 𝑏)} βˆͺ {βˆ…}) Β¬ 𝑒 ⊊ 𝑣)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ∧ wa 397   ∨ wo 846   ∧ w3a 1088  βˆ€wal 1540   = wceq 1542  βˆƒwex 1782   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  {crab 3410   βˆͺ cun 3913   ∩ cin 3914   βŠ† wss 3915   ⊊ wpss 3916  βˆ…c0 4287  π’« cpw 4565  {csn 4591  βˆͺ cuni 4870   Or wor 5549  β€˜cfv 6501   [⊊] crpss 7664  Fincfn 8890  ficfi 9353  topGenctg 17326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-rep 5247  ax-sep 5261  ax-nul 5268  ax-pow 5325  ax-pr 5389  ax-un 7677  ax-ac2 10406
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rmo 3356  df-reu 3357  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-pss 3934  df-nul 4288  df-if 4492  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-int 4913  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-tr 5228  df-id 5536  df-eprel 5542  df-po 5550  df-so 5551  df-fr 5593  df-se 5594  df-we 5595  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-pred 6258  df-ord 6325  df-on 6326  df-lim 6327  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-isom 6510  df-riota 7318  df-ov 7365  df-rpss 7665  df-om 7808  df-2nd 7927  df-frecs 8217  df-wrecs 8248  df-recs 8322  df-en 8891  df-fin 8894  df-card 9882  df-ac 10059
This theorem is referenced by:  alexsubALTlem4  23417
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