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Theorem inf3lem6 9570
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9572 for detailed description. (Contributed by NM, 29-Oct-1996.)
Hypotheses
Ref Expression
inf3lem.1 𝐺 = (𝑦 ∈ V ↦ {𝑀 ∈ π‘₯ ∣ (𝑀 ∩ π‘₯) βŠ† 𝑦})
inf3lem.2 𝐹 = (rec(𝐺, βˆ…) β†Ύ Ο‰)
inf3lem.3 𝐴 ∈ V
inf3lem.4 𝐡 ∈ V
Assertion
Ref Expression
inf3lem6 ((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) β†’ 𝐹:ω–1-1→𝒫 π‘₯)
Distinct variable group:   π‘₯,𝑦,𝑀
Allowed substitution hints:   𝐴(π‘₯,𝑦,𝑀)   𝐡(π‘₯,𝑦,𝑀)   𝐹(π‘₯,𝑦,𝑀)   𝐺(π‘₯,𝑦,𝑀)

Proof of Theorem inf3lem6
Dummy variables 𝑣 𝑒 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inf3lem.1 . . . . . . . . . . 11 𝐺 = (𝑦 ∈ V ↦ {𝑀 ∈ π‘₯ ∣ (𝑀 ∩ π‘₯) βŠ† 𝑦})
2 inf3lem.2 . . . . . . . . . . 11 𝐹 = (rec(𝐺, βˆ…) β†Ύ Ο‰)
3 vex 3450 . . . . . . . . . . 11 𝑒 ∈ V
4 vex 3450 . . . . . . . . . . 11 𝑣 ∈ V
51, 2, 3, 4inf3lem5 9569 . . . . . . . . . 10 ((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) β†’ ((𝑒 ∈ Ο‰ ∧ 𝑣 ∈ 𝑒) β†’ (πΉβ€˜π‘£) ⊊ (πΉβ€˜π‘’)))
6 dfpss2 4046 . . . . . . . . . . 11 ((πΉβ€˜π‘£) ⊊ (πΉβ€˜π‘’) ↔ ((πΉβ€˜π‘£) βŠ† (πΉβ€˜π‘’) ∧ Β¬ (πΉβ€˜π‘£) = (πΉβ€˜π‘’)))
76simprbi 498 . . . . . . . . . 10 ((πΉβ€˜π‘£) ⊊ (πΉβ€˜π‘’) β†’ Β¬ (πΉβ€˜π‘£) = (πΉβ€˜π‘’))
85, 7syl6 35 . . . . . . . . 9 ((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) β†’ ((𝑒 ∈ Ο‰ ∧ 𝑣 ∈ 𝑒) β†’ Β¬ (πΉβ€˜π‘£) = (πΉβ€˜π‘’)))
98expdimp 454 . . . . . . . 8 (((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) ∧ 𝑒 ∈ Ο‰) β†’ (𝑣 ∈ 𝑒 β†’ Β¬ (πΉβ€˜π‘£) = (πΉβ€˜π‘’)))
109adantrl 715 . . . . . . 7 (((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) ∧ (𝑣 ∈ Ο‰ ∧ 𝑒 ∈ Ο‰)) β†’ (𝑣 ∈ 𝑒 β†’ Β¬ (πΉβ€˜π‘£) = (πΉβ€˜π‘’)))
111, 2, 4, 3inf3lem5 9569 . . . . . . . . . 10 ((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) β†’ ((𝑣 ∈ Ο‰ ∧ 𝑒 ∈ 𝑣) β†’ (πΉβ€˜π‘’) ⊊ (πΉβ€˜π‘£)))
12 dfpss2 4046 . . . . . . . . . . . 12 ((πΉβ€˜π‘’) ⊊ (πΉβ€˜π‘£) ↔ ((πΉβ€˜π‘’) βŠ† (πΉβ€˜π‘£) ∧ Β¬ (πΉβ€˜π‘’) = (πΉβ€˜π‘£)))
1312simprbi 498 . . . . . . . . . . 11 ((πΉβ€˜π‘’) ⊊ (πΉβ€˜π‘£) β†’ Β¬ (πΉβ€˜π‘’) = (πΉβ€˜π‘£))
14 eqcom 2744 . . . . . . . . . . 11 ((πΉβ€˜π‘’) = (πΉβ€˜π‘£) ↔ (πΉβ€˜π‘£) = (πΉβ€˜π‘’))
1513, 14sylnib 328 . . . . . . . . . 10 ((πΉβ€˜π‘’) ⊊ (πΉβ€˜π‘£) β†’ Β¬ (πΉβ€˜π‘£) = (πΉβ€˜π‘’))
1611, 15syl6 35 . . . . . . . . 9 ((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) β†’ ((𝑣 ∈ Ο‰ ∧ 𝑒 ∈ 𝑣) β†’ Β¬ (πΉβ€˜π‘£) = (πΉβ€˜π‘’)))
1716expdimp 454 . . . . . . . 8 (((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) ∧ 𝑣 ∈ Ο‰) β†’ (𝑒 ∈ 𝑣 β†’ Β¬ (πΉβ€˜π‘£) = (πΉβ€˜π‘’)))
1817adantrr 716 . . . . . . 7 (((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) ∧ (𝑣 ∈ Ο‰ ∧ 𝑒 ∈ Ο‰)) β†’ (𝑒 ∈ 𝑣 β†’ Β¬ (πΉβ€˜π‘£) = (πΉβ€˜π‘’)))
1910, 18jaod 858 . . . . . 6 (((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) ∧ (𝑣 ∈ Ο‰ ∧ 𝑒 ∈ Ο‰)) β†’ ((𝑣 ∈ 𝑒 ∨ 𝑒 ∈ 𝑣) β†’ Β¬ (πΉβ€˜π‘£) = (πΉβ€˜π‘’)))
2019con2d 134 . . . . 5 (((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) ∧ (𝑣 ∈ Ο‰ ∧ 𝑒 ∈ Ο‰)) β†’ ((πΉβ€˜π‘£) = (πΉβ€˜π‘’) β†’ Β¬ (𝑣 ∈ 𝑒 ∨ 𝑒 ∈ 𝑣)))
21 nnord 7811 . . . . . . 7 (𝑣 ∈ Ο‰ β†’ Ord 𝑣)
22 nnord 7811 . . . . . . 7 (𝑒 ∈ Ο‰ β†’ Ord 𝑒)
23 ordtri3 6354 . . . . . . 7 ((Ord 𝑣 ∧ Ord 𝑒) β†’ (𝑣 = 𝑒 ↔ Β¬ (𝑣 ∈ 𝑒 ∨ 𝑒 ∈ 𝑣)))
2421, 22, 23syl2an 597 . . . . . 6 ((𝑣 ∈ Ο‰ ∧ 𝑒 ∈ Ο‰) β†’ (𝑣 = 𝑒 ↔ Β¬ (𝑣 ∈ 𝑒 ∨ 𝑒 ∈ 𝑣)))
2524adantl 483 . . . . 5 (((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) ∧ (𝑣 ∈ Ο‰ ∧ 𝑒 ∈ Ο‰)) β†’ (𝑣 = 𝑒 ↔ Β¬ (𝑣 ∈ 𝑒 ∨ 𝑒 ∈ 𝑣)))
2620, 25sylibrd 259 . . . 4 (((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) ∧ (𝑣 ∈ Ο‰ ∧ 𝑒 ∈ Ο‰)) β†’ ((πΉβ€˜π‘£) = (πΉβ€˜π‘’) β†’ 𝑣 = 𝑒))
2726ralrimivva 3198 . . 3 ((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) β†’ βˆ€π‘£ ∈ Ο‰ βˆ€π‘’ ∈ Ο‰ ((πΉβ€˜π‘£) = (πΉβ€˜π‘’) β†’ 𝑣 = 𝑒))
28 frfnom 8382 . . . . . 6 (rec(𝐺, βˆ…) β†Ύ Ο‰) Fn Ο‰
29 fneq1 6594 . . . . . 6 (𝐹 = (rec(𝐺, βˆ…) β†Ύ Ο‰) β†’ (𝐹 Fn Ο‰ ↔ (rec(𝐺, βˆ…) β†Ύ Ο‰) Fn Ο‰))
3028, 29mpbiri 258 . . . . 5 (𝐹 = (rec(𝐺, βˆ…) β†Ύ Ο‰) β†’ 𝐹 Fn Ο‰)
31 fvelrnb 6904 . . . . . . . 8 (𝐹 Fn Ο‰ β†’ (𝑒 ∈ ran 𝐹 ↔ βˆƒπ‘£ ∈ Ο‰ (πΉβ€˜π‘£) = 𝑒))
32 inf3lem.4 . . . . . . . . . . . 12 𝐡 ∈ V
331, 2, 4, 32inf3lemd 9564 . . . . . . . . . . 11 (𝑣 ∈ Ο‰ β†’ (πΉβ€˜π‘£) βŠ† π‘₯)
34 fvex 6856 . . . . . . . . . . . 12 (πΉβ€˜π‘£) ∈ V
3534elpw 4565 . . . . . . . . . . 11 ((πΉβ€˜π‘£) ∈ 𝒫 π‘₯ ↔ (πΉβ€˜π‘£) βŠ† π‘₯)
3633, 35sylibr 233 . . . . . . . . . 10 (𝑣 ∈ Ο‰ β†’ (πΉβ€˜π‘£) ∈ 𝒫 π‘₯)
37 eleq1 2826 . . . . . . . . . 10 ((πΉβ€˜π‘£) = 𝑒 β†’ ((πΉβ€˜π‘£) ∈ 𝒫 π‘₯ ↔ 𝑒 ∈ 𝒫 π‘₯))
3836, 37syl5ibcom 244 . . . . . . . . 9 (𝑣 ∈ Ο‰ β†’ ((πΉβ€˜π‘£) = 𝑒 β†’ 𝑒 ∈ 𝒫 π‘₯))
3938rexlimiv 3146 . . . . . . . 8 (βˆƒπ‘£ ∈ Ο‰ (πΉβ€˜π‘£) = 𝑒 β†’ 𝑒 ∈ 𝒫 π‘₯)
4031, 39syl6bi 253 . . . . . . 7 (𝐹 Fn Ο‰ β†’ (𝑒 ∈ ran 𝐹 β†’ 𝑒 ∈ 𝒫 π‘₯))
4140ssrdv 3951 . . . . . 6 (𝐹 Fn Ο‰ β†’ ran 𝐹 βŠ† 𝒫 π‘₯)
4241ancli 550 . . . . 5 (𝐹 Fn Ο‰ β†’ (𝐹 Fn Ο‰ ∧ ran 𝐹 βŠ† 𝒫 π‘₯))
432, 30, 42mp2b 10 . . . 4 (𝐹 Fn Ο‰ ∧ ran 𝐹 βŠ† 𝒫 π‘₯)
44 df-f 6501 . . . 4 (𝐹:Ο‰βŸΆπ’« π‘₯ ↔ (𝐹 Fn Ο‰ ∧ ran 𝐹 βŠ† 𝒫 π‘₯))
4543, 44mpbir 230 . . 3 𝐹:Ο‰βŸΆπ’« π‘₯
4627, 45jctil 521 . 2 ((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) β†’ (𝐹:Ο‰βŸΆπ’« π‘₯ ∧ βˆ€π‘£ ∈ Ο‰ βˆ€π‘’ ∈ Ο‰ ((πΉβ€˜π‘£) = (πΉβ€˜π‘’) β†’ 𝑣 = 𝑒)))
47 dff13 7203 . 2 (𝐹:ω–1-1→𝒫 π‘₯ ↔ (𝐹:Ο‰βŸΆπ’« π‘₯ ∧ βˆ€π‘£ ∈ Ο‰ βˆ€π‘’ ∈ Ο‰ ((πΉβ€˜π‘£) = (πΉβ€˜π‘’) β†’ 𝑣 = 𝑒)))
4846, 47sylibr 233 1 ((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) β†’ 𝐹:ω–1-1→𝒫 π‘₯)
Colors of variables: wff setvar class
Syntax hints:  Β¬ wn 3   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∨ wo 846   = wceq 1542   ∈ wcel 2107   β‰  wne 2944  βˆ€wral 3065  βˆƒwrex 3074  {crab 3408  Vcvv 3446   ∩ cin 3910   βŠ† wss 3911   ⊊ wpss 3912  βˆ…c0 4283  π’« cpw 4561  βˆͺ cuni 4866   ↦ cmpt 5189  ran crn 5635   β†Ύ cres 5636  Ord word 6317   Fn wfn 6492  βŸΆwf 6493  β€“1-1β†’wf1 6494  β€˜cfv 6497  Ο‰com 7803  reccrdg 8356
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-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673  ax-reg 9529
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-reu 3355  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-om 7804  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357
This theorem is referenced by:  inf3lem7  9571  dominf  10382  dominfac  10510
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