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Theorem grothomex 9595
Description: The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 8484). Note that our proof depends on neither the Axiom of Infinity nor Regularity. (Contributed by Mario Carneiro, 19-Apr-2013.) (New usage is discouraged.)
Assertion
Ref Expression
grothomex ω ∈ V

Proof of Theorem grothomex
Dummy variables 𝑥 𝑦 𝑧 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r111 8582 . . . 4 𝑅1:On–1-1→V
2 omsson 7016 . . . 4 ω ⊆ On
3 f1ores 6108 . . . 4 ((𝑅1:On–1-1→V ∧ ω ⊆ On) → (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω))
41, 2, 3mp2an 707 . . 3 (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω)
5 f1of1 6093 . . 3 ((𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω) → (𝑅1 ↾ ω):ω–1-1→(𝑅1 “ ω))
64, 5ax-mp 5 . 2 (𝑅1 ↾ ω):ω–1-1→(𝑅1 “ ω)
7 r1fnon 8574 . . . . . . . 8 𝑅1 Fn On
8 fvelimab 6210 . . . . . . . 8 ((𝑅1 Fn On ∧ ω ⊆ On) → (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑤))
97, 2, 8mp2an 707 . . . . . . 7 (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑤)
10 fveq2 6148 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
1110eleq1d 2683 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1‘∅) ∈ 𝑦))
12 fveq2 6148 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑅1𝑥) = (𝑅1𝑤))
1312eleq1d 2683 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1𝑤) ∈ 𝑦))
14 fveq2 6148 . . . . . . . . . . 11 (𝑥 = suc 𝑤 → (𝑅1𝑥) = (𝑅1‘suc 𝑤))
1514eleq1d 2683 . . . . . . . . . 10 (𝑥 = suc 𝑤 → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1‘suc 𝑤) ∈ 𝑦))
16 r10 8575 . . . . . . . . . . . . 13 (𝑅1‘∅) = ∅
1716eleq1i 2689 . . . . . . . . . . . 12 ((𝑅1‘∅) ∈ 𝑦 ↔ ∅ ∈ 𝑦)
1817biimpri 218 . . . . . . . . . . 11 (∅ ∈ 𝑦 → (𝑅1‘∅) ∈ 𝑦)
1918adantr 481 . . . . . . . . . 10 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1‘∅) ∈ 𝑦)
20 pweq 4133 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅1𝑤) → 𝒫 𝑧 = 𝒫 (𝑅1𝑤))
2120eleq1d 2683 . . . . . . . . . . . . . 14 (𝑧 = (𝑅1𝑤) → (𝒫 𝑧𝑦 ↔ 𝒫 (𝑅1𝑤) ∈ 𝑦))
2221rspccv 3292 . . . . . . . . . . . . 13 (∀𝑧𝑦 𝒫 𝑧𝑦 → ((𝑅1𝑤) ∈ 𝑦 → 𝒫 (𝑅1𝑤) ∈ 𝑦))
23 nnon 7018 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ω → 𝑤 ∈ On)
24 r1suc 8577 . . . . . . . . . . . . . . . 16 (𝑤 ∈ On → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1𝑤))
2523, 24syl 17 . . . . . . . . . . . . . . 15 (𝑤 ∈ ω → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1𝑤))
2625eleq1d 2683 . . . . . . . . . . . . . 14 (𝑤 ∈ ω → ((𝑅1‘suc 𝑤) ∈ 𝑦 ↔ 𝒫 (𝑅1𝑤) ∈ 𝑦))
2726biimprcd 240 . . . . . . . . . . . . 13 (𝒫 (𝑅1𝑤) ∈ 𝑦 → (𝑤 ∈ ω → (𝑅1‘suc 𝑤) ∈ 𝑦))
2822, 27syl6 35 . . . . . . . . . . . 12 (∀𝑧𝑦 𝒫 𝑧𝑦 → ((𝑅1𝑤) ∈ 𝑦 → (𝑤 ∈ ω → (𝑅1‘suc 𝑤) ∈ 𝑦)))
2928com3r 87 . . . . . . . . . . 11 (𝑤 ∈ ω → (∀𝑧𝑦 𝒫 𝑧𝑦 → ((𝑅1𝑤) ∈ 𝑦 → (𝑅1‘suc 𝑤) ∈ 𝑦)))
3029adantld 483 . . . . . . . . . 10 (𝑤 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → ((𝑅1𝑤) ∈ 𝑦 → (𝑅1‘suc 𝑤) ∈ 𝑦)))
3111, 13, 15, 19, 30finds2 7041 . . . . . . . . 9 (𝑥 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1𝑥) ∈ 𝑦))
32 eleq1 2686 . . . . . . . . . 10 ((𝑅1𝑥) = 𝑤 → ((𝑅1𝑥) ∈ 𝑦𝑤𝑦))
3332biimpd 219 . . . . . . . . 9 ((𝑅1𝑥) = 𝑤 → ((𝑅1𝑥) ∈ 𝑦𝑤𝑦))
3431, 33syl9 77 . . . . . . . 8 (𝑥 ∈ ω → ((𝑅1𝑥) = 𝑤 → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → 𝑤𝑦)))
3534rexlimiv 3020 . . . . . . 7 (∃𝑥 ∈ ω (𝑅1𝑥) = 𝑤 → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → 𝑤𝑦))
369, 35sylbi 207 . . . . . 6 (𝑤 ∈ (𝑅1 “ ω) → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → 𝑤𝑦))
3736com12 32 . . . . 5 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑤 ∈ (𝑅1 “ ω) → 𝑤𝑦))
3837ssrdv 3589 . . . 4 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1 “ ω) ⊆ 𝑦)
39 vex 3189 . . . . 5 𝑦 ∈ V
4039ssex 4762 . . . 4 ((𝑅1 “ ω) ⊆ 𝑦 → (𝑅1 “ ω) ∈ V)
4138, 40syl 17 . . 3 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1 “ ω) ∈ V)
42 0ex 4750 . . . 4 ∅ ∈ V
43 eleq1 2686 . . . . . 6 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ∈ 𝑦))
4443anbi1d 740 . . . . 5 (𝑥 = ∅ → ((𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) ↔ (∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)))
4544exbidv 1847 . . . 4 (𝑥 = ∅ → (∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) ↔ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)))
46 axgroth6 9594 . . . . 5 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
47 simpr 477 . . . . . . . 8 ((𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) → 𝒫 𝑧𝑦)
4847ralimi 2947 . . . . . . 7 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) → ∀𝑧𝑦 𝒫 𝑧𝑦)
4948anim2i 592 . . . . . 6 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦)) → (𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦))
50493adant3 1079 . . . . 5 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) → (𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦))
5146, 50eximii 1761 . . . 4 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)
5242, 45, 51vtocl 3245 . . 3 𝑦(∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)
5341, 52exlimiiv 1856 . 2 (𝑅1 “ ω) ∈ V
54 f1dmex 7083 . 2 (((𝑅1 ↾ ω):ω–1-1→(𝑅1 “ ω) ∧ (𝑅1 “ ω) ∈ V) → ω ∈ V)
556, 53, 54mp2an 707 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384  w3a 1036   = wceq 1480  wex 1701  wcel 1987  wral 2907  wrex 2908  Vcvv 3186  wss 3555  c0 3891  𝒫 cpw 4130   class class class wbr 4613  cres 5076  cima 5077  Oncon0 5682  suc csuc 5684   Fn wfn 5842  1-1wf1 5844  1-1-ontowf1o 5846  cfv 5847  ωcom 7012  csdm 7898  𝑅1cr1 8569
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902  ax-groth 9589
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-iun 4487  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-om 7013  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-er 7687  df-en 7900  df-dom 7901  df-sdom 7902  df-r1 8571
This theorem is referenced by: (None)
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