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Theorem grothomex 10245
Description: The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9099). 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 9197 . . . 4 𝑅1:On–1-1→V
2 omsson 7575 . . . 4 ω ⊆ On
3 f1ores 6618 . . . 4 ((𝑅1:On–1-1→V ∧ ω ⊆ On) → (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω))
41, 2, 3mp2an 691 . . 3 (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω)
5 f1of1 6603 . . 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 9189 . . . . . . . 8 𝑅1 Fn On
8 fvelimab 6726 . . . . . . . 8 ((𝑅1 Fn On ∧ ω ⊆ On) → (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑤))
97, 2, 8mp2an 691 . . . . . . 7 (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑤)
10 fveq2 6659 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
1110eleq1d 2900 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1‘∅) ∈ 𝑦))
12 fveq2 6659 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑅1𝑥) = (𝑅1𝑤))
1312eleq1d 2900 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1𝑤) ∈ 𝑦))
14 fveq2 6659 . . . . . . . . . . 11 (𝑥 = suc 𝑤 → (𝑅1𝑥) = (𝑅1‘suc 𝑤))
1514eleq1d 2900 . . . . . . . . . 10 (𝑥 = suc 𝑤 → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1‘suc 𝑤) ∈ 𝑦))
16 r10 9190 . . . . . . . . . . . . 13 (𝑅1‘∅) = ∅
1716eleq1i 2906 . . . . . . . . . . . 12 ((𝑅1‘∅) ∈ 𝑦 ↔ ∅ ∈ 𝑦)
1817biimpri 231 . . . . . . . . . . 11 (∅ ∈ 𝑦 → (𝑅1‘∅) ∈ 𝑦)
1918adantr 484 . . . . . . . . . 10 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1‘∅) ∈ 𝑦)
20 pweq 4538 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅1𝑤) → 𝒫 𝑧 = 𝒫 (𝑅1𝑤))
2120eleq1d 2900 . . . . . . . . . . . . . 14 (𝑧 = (𝑅1𝑤) → (𝒫 𝑧𝑦 ↔ 𝒫 (𝑅1𝑤) ∈ 𝑦))
2221rspccv 3606 . . . . . . . . . . . . 13 (∀𝑧𝑦 𝒫 𝑧𝑦 → ((𝑅1𝑤) ∈ 𝑦 → 𝒫 (𝑅1𝑤) ∈ 𝑦))
23 nnon 7577 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ω → 𝑤 ∈ On)
24 r1suc 9192 . . . . . . . . . . . . . . . 16 (𝑤 ∈ On → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1𝑤))
2523, 24syl 17 . . . . . . . . . . . . . . 15 (𝑤 ∈ ω → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1𝑤))
2625eleq1d 2900 . . . . . . . . . . . . . 14 (𝑤 ∈ ω → ((𝑅1‘suc 𝑤) ∈ 𝑦 ↔ 𝒫 (𝑅1𝑤) ∈ 𝑦))
2726biimprcd 253 . . . . . . . . . . . . 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 494 . . . . . . . . . 10 (𝑤 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → ((𝑅1𝑤) ∈ 𝑦 → (𝑅1‘suc 𝑤) ∈ 𝑦)))
3111, 13, 15, 19, 30finds2 7602 . . . . . . . . 9 (𝑥 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1𝑥) ∈ 𝑦))
32 eleq1 2903 . . . . . . . . . 10 ((𝑅1𝑥) = 𝑤 → ((𝑅1𝑥) ∈ 𝑦𝑤𝑦))
3332biimpd 232 . . . . . . . . 9 ((𝑅1𝑥) = 𝑤 → ((𝑅1𝑥) ∈ 𝑦𝑤𝑦))
3431, 33syl9 77 . . . . . . . 8 (𝑥 ∈ ω → ((𝑅1𝑥) = 𝑤 → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → 𝑤𝑦)))
3534rexlimiv 3273 . . . . . . 7 (∃𝑥 ∈ ω (𝑅1𝑥) = 𝑤 → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → 𝑤𝑦))
369, 35sylbi 220 . . . . . 6 (𝑤 ∈ (𝑅1 “ ω) → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → 𝑤𝑦))
3736com12 32 . . . . 5 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑤 ∈ (𝑅1 “ ω) → 𝑤𝑦))
3837ssrdv 3959 . . . 4 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1 “ ω) ⊆ 𝑦)
39 vex 3483 . . . . 5 𝑦 ∈ V
4039ssex 5212 . . . 4 ((𝑅1 “ ω) ⊆ 𝑦 → (𝑅1 “ ω) ∈ V)
4138, 40syl 17 . . 3 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1 “ ω) ∈ V)
42 0ex 5198 . . . 4 ∅ ∈ V
43 eleq1 2903 . . . . . 6 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ∈ 𝑦))
4443anbi1d 632 . . . . 5 (𝑥 = ∅ → ((𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) ↔ (∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)))
4544exbidv 1923 . . . 4 (𝑥 = ∅ → (∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) ↔ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)))
46 axgroth6 10244 . . . . 5 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
47 simpr 488 . . . . . . . 8 ((𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) → 𝒫 𝑧𝑦)
4847ralimi 3155 . . . . . . 7 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) → ∀𝑧𝑦 𝒫 𝑧𝑦)
4948anim2i 619 . . . . . 6 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦)) → (𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦))
50493adant3 1129 . . . . 5 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) → (𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦))
5146, 50eximii 1838 . . . 4 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)
5242, 45, 51vtocl 3545 . . 3 𝑦(∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)
5341, 52exlimiiv 1933 . 2 (𝑅1 “ ω) ∈ V
54 f1dmex 7650 . 2 (((𝑅1 ↾ ω):ω–1-1→(𝑅1 “ ω) ∧ (𝑅1 “ ω) ∈ V) → ω ∈ V)
556, 53, 54mp2an 691 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1084   = wceq 1538  wex 1781  wcel 2115  wral 3133  wrex 3134  Vcvv 3480  wss 3919  c0 4276  𝒫 cpw 4522   class class class wbr 5053  cres 5545  cima 5546  Oncon0 6179  suc csuc 6181   Fn wfn 6339  1-1wf1 6341  1-1-ontowf1o 6343  cfv 6344  ωcom 7571  csdm 8500  𝑅1cr1 9184
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 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-rep 5177  ax-sep 5190  ax-nul 5197  ax-pow 5254  ax-pr 5318  ax-un 7452  ax-groth 10239
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3015  df-ral 3138  df-rex 3139  df-reu 3140  df-rab 3142  df-v 3482  df-sbc 3759  df-csb 3867  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-pss 3938  df-nul 4277  df-if 4451  df-pw 4524  df-sn 4551  df-pr 4553  df-tp 4555  df-op 4557  df-uni 4826  df-iun 4908  df-br 5054  df-opab 5116  df-mpt 5134  df-tr 5160  df-id 5448  df-eprel 5453  df-po 5462  df-so 5463  df-fr 5502  df-we 5504  df-xp 5549  df-rel 5550  df-cnv 5551  df-co 5552  df-dm 5553  df-rn 5554  df-res 5555  df-ima 5556  df-pred 6136  df-ord 6182  df-on 6183  df-lim 6184  df-suc 6185  df-iota 6303  df-fun 6346  df-fn 6347  df-f 6348  df-f1 6349  df-fo 6350  df-f1o 6351  df-fv 6352  df-om 7572  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-er 8281  df-en 8502  df-dom 8503  df-sdom 8504  df-r1 9186
This theorem is referenced by: (None)
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