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Theorem axinf2 9678
Description: A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 9676 and Regularity ax-reg 9630.

This theorem should not be referenced in any proof. Instead, use ax-inf2 9679 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 7911 . . 3 ∅ ∈ ω
2 peano2 7913 . . . 4 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
32ax-gen 1792 . . 3 𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)
4 zfinf 9677 . . . . . 6 𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
54inf2 9661 . . . . 5 𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)
65inf3 9673 . . . 4 ω ∈ V
7 eleq2 2828 . . . . 5 (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω))
8 eleq2 2828 . . . . . . 7 (𝑥 = ω → (𝑦𝑥𝑦 ∈ ω))
9 eleq2 2828 . . . . . . 7 (𝑥 = ω → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ ω))
108, 9imbi12d 344 . . . . . 6 (𝑥 = ω → ((𝑦𝑥 → suc 𝑦𝑥) ↔ (𝑦 ∈ ω → suc 𝑦 ∈ ω)))
1110albidv 1918 . . . . 5 (𝑥 = ω → (∀𝑦(𝑦𝑥 → suc 𝑦𝑥) ↔ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)))
127, 11anbi12d 632 . . . 4 (𝑥 = ω → ((∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)) ↔ (∅ ∈ ω ∧ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω))))
136, 12spcev 3606 . . 3 ((∅ ∈ ω ∧ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)) → ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)))
141, 3, 13mp2an 692 . 2 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥))
15 0el 4369 . . . . 5 (∅ ∈ 𝑥 ↔ ∃𝑦𝑥𝑧 ¬ 𝑧𝑦)
16 df-rex 3069 . . . . 5 (∃𝑦𝑥𝑧 ¬ 𝑧𝑦 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
1715, 16bitri 275 . . . 4 (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
18 sucel 6460 . . . . . . 7 (suc 𝑦𝑥 ↔ ∃𝑧𝑥𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))
19 df-rex 3069 . . . . . . 7 (∃𝑧𝑥𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)) ↔ ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
2018, 19bitri 275 . . . . . 6 (suc 𝑦𝑥 ↔ ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
2120imbi2i 336 . . . . 5 ((𝑦𝑥 → suc 𝑦𝑥) ↔ (𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
2221albii 1816 . . . 4 (∀𝑦(𝑦𝑥 → suc 𝑦𝑥) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
2317, 22anbi12i 628 . . 3 ((∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)) ↔ (∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))))
2423exbii 1845 . 2 (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)) ↔ ∃𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))))
2514, 24mpbi 230 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 206  wa 395  wo 847  wal 1535   = wceq 1537  wex 1776  wcel 2106  wrex 3068  c0 4339  suc csuc 6388  ωcom 7887
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pow 5371  ax-pr 5438  ax-un 7754  ax-reg 9630  ax-inf 9676
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-pred 6323  df-ord 6389  df-on 6390  df-lim 6391  df-suc 6392  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-om 7888  df-2nd 8014  df-frecs 8305  df-wrecs 8336  df-recs 8410  df-rdg 8449
This theorem is referenced by: (None)
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