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Theorem grothomex 9853
Description: The Tarski-Grothendieck Axiom implies the Axiom of Infinity (in the form of omex 8704). 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 8802 . . . 4 𝑅1:On–1-1→V
2 omsson 7216 . . . 4 ω ⊆ On
3 f1ores 6292 . . . 4 ((𝑅1:On–1-1→V ∧ ω ⊆ On) → (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω))
41, 2, 3mp2an 672 . . 3 (𝑅1 ↾ ω):ω–1-1-onto→(𝑅1 “ ω)
5 f1of1 6277 . . 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 8794 . . . . . . . 8 𝑅1 Fn On
8 fvelimab 6395 . . . . . . . 8 ((𝑅1 Fn On ∧ ω ⊆ On) → (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑤))
97, 2, 8mp2an 672 . . . . . . 7 (𝑤 ∈ (𝑅1 “ ω) ↔ ∃𝑥 ∈ ω (𝑅1𝑥) = 𝑤)
10 fveq2 6332 . . . . . . . . . . 11 (𝑥 = ∅ → (𝑅1𝑥) = (𝑅1‘∅))
1110eleq1d 2835 . . . . . . . . . 10 (𝑥 = ∅ → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1‘∅) ∈ 𝑦))
12 fveq2 6332 . . . . . . . . . . 11 (𝑥 = 𝑤 → (𝑅1𝑥) = (𝑅1𝑤))
1312eleq1d 2835 . . . . . . . . . 10 (𝑥 = 𝑤 → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1𝑤) ∈ 𝑦))
14 fveq2 6332 . . . . . . . . . . 11 (𝑥 = suc 𝑤 → (𝑅1𝑥) = (𝑅1‘suc 𝑤))
1514eleq1d 2835 . . . . . . . . . 10 (𝑥 = suc 𝑤 → ((𝑅1𝑥) ∈ 𝑦 ↔ (𝑅1‘suc 𝑤) ∈ 𝑦))
16 r10 8795 . . . . . . . . . . . . 13 (𝑅1‘∅) = ∅
1716eleq1i 2841 . . . . . . . . . . . 12 ((𝑅1‘∅) ∈ 𝑦 ↔ ∅ ∈ 𝑦)
1817biimpri 218 . . . . . . . . . . 11 (∅ ∈ 𝑦 → (𝑅1‘∅) ∈ 𝑦)
1918adantr 466 . . . . . . . . . 10 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1‘∅) ∈ 𝑦)
20 pweq 4300 . . . . . . . . . . . . . . 15 (𝑧 = (𝑅1𝑤) → 𝒫 𝑧 = 𝒫 (𝑅1𝑤))
2120eleq1d 2835 . . . . . . . . . . . . . 14 (𝑧 = (𝑅1𝑤) → (𝒫 𝑧𝑦 ↔ 𝒫 (𝑅1𝑤) ∈ 𝑦))
2221rspccv 3457 . . . . . . . . . . . . 13 (∀𝑧𝑦 𝒫 𝑧𝑦 → ((𝑅1𝑤) ∈ 𝑦 → 𝒫 (𝑅1𝑤) ∈ 𝑦))
23 nnon 7218 . . . . . . . . . . . . . . . 16 (𝑤 ∈ ω → 𝑤 ∈ On)
24 r1suc 8797 . . . . . . . . . . . . . . . 16 (𝑤 ∈ On → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1𝑤))
2523, 24syl 17 . . . . . . . . . . . . . . 15 (𝑤 ∈ ω → (𝑅1‘suc 𝑤) = 𝒫 (𝑅1𝑤))
2625eleq1d 2835 . . . . . . . . . . . . . 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 478 . . . . . . . . . 10 (𝑤 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → ((𝑅1𝑤) ∈ 𝑦 → (𝑅1‘suc 𝑤) ∈ 𝑦)))
3111, 13, 15, 19, 30finds2 7241 . . . . . . . . 9 (𝑥 ∈ ω → ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1𝑥) ∈ 𝑦))
32 eleq1 2838 . . . . . . . . . 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 3175 . . . . . . 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 3758 . . . 4 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1 “ ω) ⊆ 𝑦)
39 vex 3354 . . . . 5 𝑦 ∈ V
4039ssex 4936 . . . 4 ((𝑅1 “ ω) ⊆ 𝑦 → (𝑅1 “ ω) ∈ V)
4138, 40syl 17 . . 3 ((∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) → (𝑅1 “ ω) ∈ V)
42 0ex 4924 . . . 4 ∅ ∈ V
43 eleq1 2838 . . . . . 6 (𝑥 = ∅ → (𝑥𝑦 ↔ ∅ ∈ 𝑦))
4443anbi1d 615 . . . . 5 (𝑥 = ∅ → ((𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) ↔ (∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)))
4544exbidv 2002 . . . 4 (𝑥 = ∅ → (∃𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦) ↔ ∃𝑦(∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)))
46 axgroth6 9852 . . . . 5 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦))
47 simpr 471 . . . . . . . 8 ((𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) → 𝒫 𝑧𝑦)
4847ralimi 3101 . . . . . . 7 (∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) → ∀𝑧𝑦 𝒫 𝑧𝑦)
4948anim2i 603 . . . . . 6 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦)) → (𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦))
50493adant3 1126 . . . . 5 ((𝑥𝑦 ∧ ∀𝑧𝑦 (𝒫 𝑧𝑦 ∧ 𝒫 𝑧𝑦) ∧ ∀𝑧 ∈ 𝒫 𝑦(𝑧𝑦𝑧𝑦)) → (𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦))
5146, 50eximii 1912 . . . 4 𝑦(𝑥𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)
5242, 45, 51vtocl 3410 . . 3 𝑦(∅ ∈ 𝑦 ∧ ∀𝑧𝑦 𝒫 𝑧𝑦)
5341, 52exlimiiv 2011 . 2 (𝑅1 “ ω) ∈ V
54 f1dmex 7283 . 2 (((𝑅1 ↾ ω):ω–1-1→(𝑅1 “ ω) ∧ (𝑅1 “ ω) ∈ V) → ω ∈ V)
556, 53, 54mp2an 672 1 ω ∈ V
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  wral 3061  wrex 3062  Vcvv 3351  wss 3723  c0 4063  𝒫 cpw 4297   class class class wbr 4786  cres 5251  cima 5252  Oncon0 5866  suc csuc 5868   Fn wfn 6026  1-1wf1 6028  1-1-ontowf1o 6030  cfv 6031  ωcom 7212  csdm 8108  𝑅1cr1 8789
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-groth 9847
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-er 7896  df-en 8110  df-dom 8111  df-sdom 8112  df-r1 8791
This theorem is referenced by: (None)
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