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Theorem inf3lem6 9630
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 9632 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 3478 . . . . . . . . . . 11 𝑒 ∈ V
4 vex 3478 . . . . . . . . . . 11 𝑣 ∈ V
51, 2, 3, 4inf3lem5 9629 . . . . . . . . . 10 ((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) β†’ ((𝑒 ∈ Ο‰ ∧ 𝑣 ∈ 𝑒) β†’ (πΉβ€˜π‘£) ⊊ (πΉβ€˜π‘’)))
6 dfpss2 4085 . . . . . . . . . . 11 ((πΉβ€˜π‘£) ⊊ (πΉβ€˜π‘’) ↔ ((πΉβ€˜π‘£) βŠ† (πΉβ€˜π‘’) ∧ Β¬ (πΉβ€˜π‘£) = (πΉβ€˜π‘’)))
76simprbi 497 . . . . . . . . . 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 453 . . . . . . . 8 (((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) ∧ 𝑒 ∈ Ο‰) β†’ (𝑣 ∈ 𝑒 β†’ Β¬ (πΉβ€˜π‘£) = (πΉβ€˜π‘’)))
109adantrl 714 . . . . . . 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 9629 . . . . . . . . . 10 ((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) β†’ ((𝑣 ∈ Ο‰ ∧ 𝑒 ∈ 𝑣) β†’ (πΉβ€˜π‘’) ⊊ (πΉβ€˜π‘£)))
12 dfpss2 4085 . . . . . . . . . . . 12 ((πΉβ€˜π‘’) ⊊ (πΉβ€˜π‘£) ↔ ((πΉβ€˜π‘’) βŠ† (πΉβ€˜π‘£) ∧ Β¬ (πΉβ€˜π‘’) = (πΉβ€˜π‘£)))
1312simprbi 497 . . . . . . . . . . 11 ((πΉβ€˜π‘’) ⊊ (πΉβ€˜π‘£) β†’ Β¬ (πΉβ€˜π‘’) = (πΉβ€˜π‘£))
14 eqcom 2739 . . . . . . . . . . 11 ((πΉβ€˜π‘’) = (πΉβ€˜π‘£) ↔ (πΉβ€˜π‘£) = (πΉβ€˜π‘’))
1513, 14sylnib 327 . . . . . . . . . 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 453 . . . . . . . 8 (((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) ∧ 𝑣 ∈ Ο‰) β†’ (𝑒 ∈ 𝑣 β†’ Β¬ (πΉβ€˜π‘£) = (πΉβ€˜π‘’)))
1817adantrr 715 . . . . . . 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 857 . . . . . 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 7865 . . . . . . 7 (𝑣 ∈ Ο‰ β†’ Ord 𝑣)
22 nnord 7865 . . . . . . 7 (𝑒 ∈ Ο‰ β†’ Ord 𝑒)
23 ordtri3 6400 . . . . . . 7 ((Ord 𝑣 ∧ Ord 𝑒) β†’ (𝑣 = 𝑒 ↔ Β¬ (𝑣 ∈ 𝑒 ∨ 𝑒 ∈ 𝑣)))
2421, 22, 23syl2an 596 . . . . . 6 ((𝑣 ∈ Ο‰ ∧ 𝑒 ∈ Ο‰) β†’ (𝑣 = 𝑒 ↔ Β¬ (𝑣 ∈ 𝑒 ∨ 𝑒 ∈ 𝑣)))
2524adantl 482 . . . . 5 (((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) ∧ (𝑣 ∈ Ο‰ ∧ 𝑒 ∈ Ο‰)) β†’ (𝑣 = 𝑒 ↔ Β¬ (𝑣 ∈ 𝑒 ∨ 𝑒 ∈ 𝑣)))
2620, 25sylibrd 258 . . . 4 (((π‘₯ β‰  βˆ… ∧ π‘₯ βŠ† βˆͺ π‘₯) ∧ (𝑣 ∈ Ο‰ ∧ 𝑒 ∈ Ο‰)) β†’ ((πΉβ€˜π‘£) = (πΉβ€˜π‘’) β†’ 𝑣 = 𝑒))
2726ralrimivva 3200 . . 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 8437 . . . . . 6 (rec(𝐺, βˆ…) β†Ύ Ο‰) Fn Ο‰
29 fneq1 6640 . . . . . 6 (𝐹 = (rec(𝐺, βˆ…) β†Ύ Ο‰) β†’ (𝐹 Fn Ο‰ ↔ (rec(𝐺, βˆ…) β†Ύ Ο‰) Fn Ο‰))
3028, 29mpbiri 257 . . . . 5 (𝐹 = (rec(𝐺, βˆ…) β†Ύ Ο‰) β†’ 𝐹 Fn Ο‰)
31 fvelrnb 6952 . . . . . . . 8 (𝐹 Fn Ο‰ β†’ (𝑒 ∈ ran 𝐹 ↔ βˆƒπ‘£ ∈ Ο‰ (πΉβ€˜π‘£) = 𝑒))
32 inf3lem.4 . . . . . . . . . . . 12 𝐡 ∈ V
331, 2, 4, 32inf3lemd 9624 . . . . . . . . . . 11 (𝑣 ∈ Ο‰ β†’ (πΉβ€˜π‘£) βŠ† π‘₯)
34 fvex 6904 . . . . . . . . . . . 12 (πΉβ€˜π‘£) ∈ V
3534elpw 4606 . . . . . . . . . . 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 2821 . . . . . . . . . 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 3148 . . . . . . . 8 (βˆƒπ‘£ ∈ Ο‰ (πΉβ€˜π‘£) = 𝑒 β†’ 𝑒 ∈ 𝒫 π‘₯)
4031, 39syl6bi 252 . . . . . . 7 (𝐹 Fn Ο‰ β†’ (𝑒 ∈ ran 𝐹 β†’ 𝑒 ∈ 𝒫 π‘₯))
4140ssrdv 3988 . . . . . 6 (𝐹 Fn Ο‰ β†’ ran 𝐹 βŠ† 𝒫 π‘₯)
4241ancli 549 . . . . 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 6547 . . . 4 (𝐹:Ο‰βŸΆπ’« π‘₯ ↔ (𝐹 Fn Ο‰ ∧ ran 𝐹 βŠ† 𝒫 π‘₯))
4543, 44mpbir 230 . . 3 𝐹:Ο‰βŸΆπ’« π‘₯
4627, 45jctil 520 . 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 7256 . 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 396   ∨ wo 845   = wceq 1541   ∈ wcel 2106   β‰  wne 2940  βˆ€wral 3061  βˆƒwrex 3070  {crab 3432  Vcvv 3474   ∩ cin 3947   βŠ† wss 3948   ⊊ wpss 3949  βˆ…c0 4322  π’« cpw 4602  βˆͺ cuni 4908   ↦ cmpt 5231  ran crn 5677   β†Ύ cres 5678  Ord word 6363   Fn wfn 6538  βŸΆwf 6539  β€“1-1β†’wf1 6540  β€˜cfv 6543  Ο‰com 7857  reccrdg 8411
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727  ax-reg 9589
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6300  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7414  df-om 7858  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412
This theorem is referenced by:  inf3lem7  9631  dominf  10442  dominfac  10570
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