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Theorem grothomex 10765
Description: The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 9579). 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 9711 . . . 4 𝑅1:On–1-1→V
2 omsson 7806 . . . 4 ω ⊆ On
3 f1ores 6798 . . . 4 ((𝑅1:On–1-1→V ∧ ω ⊆ On) → (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω))
41, 2, 3mp2an 690 . . 3 (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω)
5 f1of1 6783 . . 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 9703 . . . . . . . 8 𝑅1 Fn On
8 fvelimab 6914 . . . . . . . 8 ((𝑅1 Fn On ∧ ω ⊆ On) → (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑤))
97, 2, 8mp2an 690 . . . . . . 7 (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑤)
10 fveq2 6842 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
1110eleq1d 2822 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1‘∅) ∈ 𝑦))
12 fveq2 6842 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑅1𝑥) = (𝑅1𝑤))
1312eleq1d 2822 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1𝑤) ∈ 𝑦))
14 fveq2 6842 . . . . . . . . . . 11 (𝑥 = suc 𝑤 → (𝑅1𝑥) = (𝑅1‘suc 𝑤))
1514eleq1d 2822 . . . . . . . . . 10 (𝑥 = suc 𝑤 → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1‘suc 𝑤) ∈ 𝑦))
16 r10 9704 . . . . . . . . . . . . 13 (𝑅1‘∅) = ∅
1716eleq1i 2828 . . . . . . . . . . . 12 ((𝑅1‘∅) ∈ 𝑦 ↔ ∅ ∈ 𝑦)
1817biimpri 227 . . . . . . . . . . 11 (∅ ∈ 𝑦 → (𝑅1‘∅) ∈ 𝑦)
1918adantr 481 . . . . . . . . . 10 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1‘∅) ∈ 𝑦)
20 pweq 4574 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅1𝑤) → 𝒫 𝑧 = 𝒫 (𝑅1𝑤))
2120eleq1d 2822 . . . . . . . . . . . . . 14 (𝑧 = (𝑅1𝑤) → (𝒫 𝑧𝑦 ↔ 𝒫 (𝑅1𝑤) ∈ 𝑦))
2221rspccv 3578 . . . . . . . . . . . . 13 (∀𝑧𝑦 𝒫 𝑧𝑦 → ((𝑅1𝑤) ∈ 𝑦 → 𝒫 (𝑅1𝑤) ∈ 𝑦))
23 nnon 7808 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ω → 𝑤 ∈ On)
24 r1suc 9706 . . . . . . . . . . . . . . . 16 (𝑤 ∈ On → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1𝑤))
2523, 24syl 17 . . . . . . . . . . . . . . 15 (𝑤 ∈ ω → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1𝑤))
2625eleq1d 2822 . . . . . . . . . . . . . 14 (𝑤 ∈ ω → ((𝑅1‘suc 𝑤) ∈ 𝑦 ↔ 𝒫 (𝑅1𝑤) ∈ 𝑦))
2726biimprcd 249 . . . . . . . . . . . . 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 491 . . . . . . . . . 10 (𝑤 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → ((𝑅1𝑤) ∈ 𝑦 → (𝑅1‘suc 𝑤) ∈ 𝑦)))
3111, 13, 15, 19, 30finds2 7837 . . . . . . . . 9 (𝑥 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1𝑥) ∈ 𝑦))
32 eleq1 2825 . . . . . . . . . 10 ((𝑅1𝑥) = 𝑤 → ((𝑅1𝑥) ∈ 𝑦𝑤𝑦))
3332biimpd 228 . . . . . . . . 9 ((𝑅1𝑥) = 𝑤 → ((𝑅1𝑥) ∈ 𝑦𝑤𝑦))
3431, 33syl9 77 . . . . . . . 8 (𝑥 ∈ ω → ((𝑅1𝑥) = 𝑤 → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → 𝑤𝑦)))
3534rexlimiv 3145 . . . . . . 7 (∃𝑥 ∈ ω (𝑅1𝑥) = 𝑤 → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → 𝑤𝑦))
369, 35sylbi 216 . . . . . 6 (𝑤 ∈ (𝑅1 “ ω) → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → 𝑤𝑦))
3736com12 32 . . . . 5 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑤 ∈ (𝑅1 “ ω) → 𝑤𝑦))
3837ssrdv 3950 . . . 4 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1 “ ω) ⊆ 𝑦)
39 vex 3449 . . . . 5 𝑦 ∈ V
4039ssex 5278 . . . 4 ((𝑅1 “ ω) ⊆ 𝑦 → (𝑅1 “ ω) ∈ V)
4138, 40syl 17 . . 3 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1 “ ω) ∈ V)
42 0ex 5264 . . . 4 ∅ ∈ V
43 eleq1 2825 . . . . . 6 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ∈ 𝑦))
4443anbi1d 630 . . . . 5 (𝑥 = ∅ → ((𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) ↔ (∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)))
4544exbidv 1924 . . . 4 (𝑥 = ∅ → (∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) ↔ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)))
46 axgroth6 10764 . . . . 5 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
47 simpr 485 . . . . . . . 8 ((𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) → 𝒫 𝑧𝑦)
4847ralimi 3086 . . . . . . 7 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) → ∀𝑧𝑦 𝒫 𝑧𝑦)
4948anim2i 617 . . . . . 6 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦)) → (𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦))
50493adant3 1132 . . . . 5 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) → (𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦))
5146, 50eximii 1839 . . . 4 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)
5242, 45, 51vtocl 3518 . . 3 𝑦(∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)
5341, 52exlimiiv 1934 . 2 (𝑅1 “ ω) ∈ V
54 f1dmex 7889 . 2 (((𝑅1 ↾ ω):ω–1-1→(𝑅1 “ ω) ∧ (𝑅1 “ ω) ∈ V) → ω ∈ V)
556, 53, 54mp2an 690 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1087   = wceq 1541  wex 1781  wcel 2106  wral 3064  wrex 3073  Vcvv 3445  wss 3910  c0 4282  𝒫 cpw 4560   class class class wbr 5105  cres 5635  cima 5636  Oncon0 6317  suc csuc 6319   Fn wfn 6491  1-1wf1 6493  1-1-ontowf1o 6495  cfv 6496  ωcom 7802  csdm 8882  𝑅1cr1 9698
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 2707  ax-rep 5242  ax-sep 5256  ax-nul 5263  ax-pow 5320  ax-pr 5384  ax-un 7672  ax-groth 10759
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 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-reu 3354  df-rab 3408  df-v 3447  df-sbc 3740  df-csb 3856  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-pss 3929  df-nul 4283  df-if 4487  df-pw 4562  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-iun 4956  df-br 5106  df-opab 5168  df-mpt 5189  df-tr 5223  df-id 5531  df-eprel 5537  df-po 5545  df-so 5546  df-fr 5588  df-we 5590  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-pred 6253  df-ord 6320  df-on 6321  df-lim 6322  df-suc 6323  df-iota 6448  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-fv 6504  df-ov 7360  df-om 7803  df-2nd 7922  df-frecs 8212  df-wrecs 8243  df-recs 8317  df-rdg 8356  df-er 8648  df-en 8884  df-dom 8885  df-sdom 8886  df-r1 9700
This theorem is referenced by: (None)
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