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Theorem axinf2 9676
Description: A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 9674 and Regularity ax-reg 9628.

This theorem should not be referenced in any proof. Instead, use ax-inf2 9677 below so that the ordinary uses of Regularity can be more easily identified. (New usage is discouraged.) (Contributed by NM, 3-Nov-1996.)

Assertion
Ref Expression
axinf2 𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem axinf2
StepHypRef Expression
1 peano1 7892 . . 3 ∅ ∈ ω
2 peano2 7894 . . . 4 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
32ax-gen 1790 . . 3 𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)
4 zfinf 9675 . . . . . 6 𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
54inf2 9659 . . . . 5 𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)
65inf3 9671 . . . 4 ω ∈ V
7 eleq2 2815 . . . . 5 (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω))
8 eleq2 2815 . . . . . . 7 (𝑥 = ω → (𝑦𝑥𝑦 ∈ ω))
9 eleq2 2815 . . . . . . 7 (𝑥 = ω → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ ω))
108, 9imbi12d 343 . . . . . 6 (𝑥 = ω → ((𝑦𝑥 → suc 𝑦𝑥) ↔ (𝑦 ∈ ω → suc 𝑦 ∈ ω)))
1110albidv 1916 . . . . 5 (𝑥 = ω → (∀𝑦(𝑦𝑥 → suc 𝑦𝑥) ↔ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)))
127, 11anbi12d 630 . . . 4 (𝑥 = ω → ((∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)) ↔ (∅ ∈ ω ∧ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω))))
136, 12spcev 3591 . . 3 ((∅ ∈ ω ∧ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)) → ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)))
141, 3, 13mp2an 690 . 2 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥))
15 0el 4356 . . . . 5 (∅ ∈ 𝑥 ↔ ∃𝑦𝑥𝑧 ¬ 𝑧𝑦)
16 df-rex 3061 . . . . 5 (∃𝑦𝑥𝑧 ¬ 𝑧𝑦 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
1715, 16bitri 274 . . . 4 (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
18 sucel 6442 . . . . . . 7 (suc 𝑦𝑥 ↔ ∃𝑧𝑥𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))
19 df-rex 3061 . . . . . . 7 (∃𝑧𝑥𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)) ↔ ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
2018, 19bitri 274 . . . . . 6 (suc 𝑦𝑥 ↔ ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
2120imbi2i 335 . . . . 5 ((𝑦𝑥 → suc 𝑦𝑥) ↔ (𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
2221albii 1814 . . . 4 (∀𝑦(𝑦𝑥 → suc 𝑦𝑥) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
2317, 22anbi12i 626 . . 3 ((∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)) ↔ (∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))))
2423exbii 1843 . 2 (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)) ↔ ∃𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))))
2514, 24mpbi 229 1 𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 845  wal 1532   = wceq 1534  wex 1774  wcel 2099  wrex 3060  c0 4322  suc csuc 6370  ωcom 7868
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5282  ax-sep 5296  ax-nul 5303  ax-pow 5361  ax-pr 5425  ax-un 7738  ax-reg 9628  ax-inf 9674
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3776  df-csb 3892  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-pss 3966  df-nul 4323  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4906  df-iun 4995  df-br 5146  df-opab 5208  df-mpt 5229  df-tr 5263  df-id 5572  df-eprel 5578  df-po 5586  df-so 5587  df-fr 5629  df-we 5631  df-xp 5680  df-rel 5681  df-cnv 5682  df-co 5683  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687  df-pred 6304  df-ord 6371  df-on 6372  df-lim 6373  df-suc 6374  df-iota 6498  df-fun 6548  df-fn 6549  df-f 6550  df-f1 6551  df-fo 6552  df-f1o 6553  df-fv 6554  df-ov 7419  df-om 7869  df-2nd 7996  df-frecs 8288  df-wrecs 8319  df-recs 8393  df-rdg 8432
This theorem is referenced by: (None)
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