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Theorem grothomex 9814
Description: The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 8701). 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 8799 . . . 4 𝑅1:On–1-1→V
2 omsson 7222 . . . 4 ω ⊆ On
3 f1ores 6300 . . . 4 ((𝑅1:On–1-1→V ∧ ω ⊆ On) → (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω))
41, 2, 3mp2an 710 . . 3 (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω)
5 f1of1 6285 . . 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 8791 . . . . . . . 8 𝑅1 Fn On
8 fvelimab 6403 . . . . . . . 8 ((𝑅1 Fn On ∧ ω ⊆ On) → (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑤))
97, 2, 8mp2an 710 . . . . . . 7 (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑤)
10 fveq2 6340 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
1110eleq1d 2812 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1‘∅) ∈ 𝑦))
12 fveq2 6340 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑅1𝑥) = (𝑅1𝑤))
1312eleq1d 2812 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1𝑤) ∈ 𝑦))
14 fveq2 6340 . . . . . . . . . . 11 (𝑥 = suc 𝑤 → (𝑅1𝑥) = (𝑅1‘suc 𝑤))
1514eleq1d 2812 . . . . . . . . . 10 (𝑥 = suc 𝑤 → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1‘suc 𝑤) ∈ 𝑦))
16 r10 8792 . . . . . . . . . . . . 13 (𝑅1‘∅) = ∅
1716eleq1i 2818 . . . . . . . . . . . 12 ((𝑅1‘∅) ∈ 𝑦 ↔ ∅ ∈ 𝑦)
1817biimpri 218 . . . . . . . . . . 11 (∅ ∈ 𝑦 → (𝑅1‘∅) ∈ 𝑦)
1918adantr 472 . . . . . . . . . 10 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1‘∅) ∈ 𝑦)
20 pweq 4293 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅1𝑤) → 𝒫 𝑧 = 𝒫 (𝑅1𝑤))
2120eleq1d 2812 . . . . . . . . . . . . . 14 (𝑧 = (𝑅1𝑤) → (𝒫 𝑧𝑦 ↔ 𝒫 (𝑅1𝑤) ∈ 𝑦))
2221rspccv 3434 . . . . . . . . . . . . 13 (∀𝑧𝑦 𝒫 𝑧𝑦 → ((𝑅1𝑤) ∈ 𝑦 → 𝒫 (𝑅1𝑤) ∈ 𝑦))
23 nnon 7224 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ω → 𝑤 ∈ On)
24 r1suc 8794 . . . . . . . . . . . . . . . 16 (𝑤 ∈ On → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1𝑤))
2523, 24syl 17 . . . . . . . . . . . . . . 15 (𝑤 ∈ ω → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1𝑤))
2625eleq1d 2812 . . . . . . . . . . . . . 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 484 . . . . . . . . . 10 (𝑤 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → ((𝑅1𝑤) ∈ 𝑦 → (𝑅1‘suc 𝑤) ∈ 𝑦)))
3111, 13, 15, 19, 30finds2 7247 . . . . . . . . 9 (𝑥 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1𝑥) ∈ 𝑦))
32 eleq1 2815 . . . . . . . . . 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 3153 . . . . . . 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 3738 . . . 4 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1 “ ω) ⊆ 𝑦)
39 vex 3331 . . . . 5 𝑦 ∈ V
4039ssex 4942 . . . 4 ((𝑅1 “ ω) ⊆ 𝑦 → (𝑅1 “ ω) ∈ V)
4138, 40syl 17 . . 3 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1 “ ω) ∈ V)
42 0ex 4930 . . . 4 ∅ ∈ V
43 eleq1 2815 . . . . . 6 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ∈ 𝑦))
4443anbi1d 743 . . . . 5 (𝑥 = ∅ → ((𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) ↔ (∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)))
4544exbidv 1987 . . . 4 (𝑥 = ∅ → (∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) ↔ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)))
46 axgroth6 9813 . . . . 5 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
47 simpr 479 . . . . . . . 8 ((𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) → 𝒫 𝑧𝑦)
4847ralimi 3078 . . . . . . 7 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) → ∀𝑧𝑦 𝒫 𝑧𝑦)
4948anim2i 594 . . . . . 6 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦)) → (𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦))
50493adant3 1124 . . . . 5 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) → (𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦))
5146, 50eximii 1901 . . . 4 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)
5242, 45, 51vtocl 3387 . . 3 𝑦(∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)
5341, 52exlimiiv 1996 . 2 (𝑅1 “ ω) ∈ V
54 f1dmex 7289 . 2 (((𝑅1 ↾ ω):ω–1-1→(𝑅1 “ ω) ∧ (𝑅1 “ ω) ∈ V) → ω ∈ V)
556, 53, 54mp2an 710 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  w3a 1072   = wceq 1620  wex 1841  wcel 2127  wral 3038  wrex 3039  Vcvv 3328  wss 3703  c0 4046  𝒫 cpw 4290   class class class wbr 4792  cres 5256  cima 5257  Oncon0 5872  suc csuc 5874   Fn wfn 6032  1-1wf1 6034  1-1-ontowf1o 6036  cfv 6037  ωcom 7218  csdm 8108  𝑅1cr1 8786
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-8 2129  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-rep 4911  ax-sep 4921  ax-nul 4929  ax-pow 4980  ax-pr 5043  ax-un 7102  ax-groth 9808
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ne 2921  df-ral 3043  df-rex 3044  df-reu 3045  df-rab 3047  df-v 3330  df-sbc 3565  df-csb 3663  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-pss 3719  df-nul 4047  df-if 4219  df-pw 4292  df-sn 4310  df-pr 4312  df-tp 4314  df-op 4316  df-uni 4577  df-iun 4662  df-br 4793  df-opab 4853  df-mpt 4870  df-tr 4893  df-id 5162  df-eprel 5167  df-po 5175  df-so 5176  df-fr 5213  df-we 5215  df-xp 5260  df-rel 5261  df-cnv 5262  df-co 5263  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-pred 5829  df-ord 5875  df-on 5876  df-lim 5877  df-suc 5878  df-iota 6000  df-fun 6039  df-fn 6040  df-f 6041  df-f1 6042  df-fo 6043  df-f1o 6044  df-fv 6045  df-om 7219  df-wrecs 7564  df-recs 7625  df-rdg 7663  df-er 7899  df-en 8110  df-dom 8111  df-sdom 8112  df-r1 8788
This theorem is referenced by: (None)
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