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Theorem axinf2 9584
Description: A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 9582 and Regularity ax-reg 9536.

This theorem should not be referenced in any proof. Instead, use ax-inf2 9585 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 7829 . . 3 ∅ ∈ ω
2 peano2 7831 . . . 4 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
32ax-gen 1798 . . 3 𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)
4 zfinf 9583 . . . . . 6 𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
54inf2 9567 . . . . 5 𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)
65inf3 9579 . . . 4 ω ∈ V
7 eleq2 2823 . . . . 5 (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω))
8 eleq2 2823 . . . . . . 7 (𝑥 = ω → (𝑦𝑥𝑦 ∈ ω))
9 eleq2 2823 . . . . . . 7 (𝑥 = ω → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ ω))
108, 9imbi12d 345 . . . . . 6 (𝑥 = ω → ((𝑦𝑥 → suc 𝑦𝑥) ↔ (𝑦 ∈ ω → suc 𝑦 ∈ ω)))
1110albidv 1924 . . . . 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 3567 . . 3 ((∅ ∈ ω ∧ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)) → ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)))
141, 3, 13mp2an 691 . 2 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥))
15 0el 4324 . . . . 5 (∅ ∈ 𝑥 ↔ ∃𝑦𝑥𝑧 ¬ 𝑧𝑦)
16 df-rex 3071 . . . . 5 (∃𝑦𝑥𝑧 ¬ 𝑧𝑦 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
1715, 16bitri 275 . . . 4 (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
18 sucel 6395 . . . . . . 7 (suc 𝑦𝑥 ↔ ∃𝑧𝑥𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))
19 df-rex 3071 . . . . . . 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 1822 . . . 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 1851 . 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 397  wo 846  wal 1540   = wceq 1542  wex 1782  wcel 2107  wrex 3070  c0 4286  suc csuc 6323  ωcom 7806
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-reg 9536  ax-inf 9582
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360
This theorem is referenced by: (None)
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