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Theorem axinf2 8534
 Description: A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 8532 and Regularity ax-reg 8494. This theorem should not be referenced in any proof. Instead, use ax-inf2 8535 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 7082 . . 3 ∅ ∈ ω
2 peano2 7083 . . . 4 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
32ax-gen 1721 . . 3 𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)
4 zfinf 8533 . . . . . 6 𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
54inf2 8517 . . . . 5 𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)
65inf3 8529 . . . 4 ω ∈ V
7 eleq2 2689 . . . . 5 (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω))
8 eleq2 2689 . . . . . . 7 (𝑥 = ω → (𝑦𝑥𝑦 ∈ ω))
9 eleq2 2689 . . . . . . 7 (𝑥 = ω → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ ω))
108, 9imbi12d 334 . . . . . 6 (𝑥 = ω → ((𝑦𝑥 → suc 𝑦𝑥) ↔ (𝑦 ∈ ω → suc 𝑦 ∈ ω)))
1110albidv 1848 . . . . 5 (𝑥 = ω → (∀𝑦(𝑦𝑥 → suc 𝑦𝑥) ↔ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)))
127, 11anbi12d 747 . . . 4 (𝑥 = ω → ((∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)) ↔ (∅ ∈ ω ∧ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω))))
136, 12spcev 3298 . . 3 ((∅ ∈ ω ∧ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)) → ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)))
141, 3, 13mp2an 708 . 2 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥))
15 0el 3937 . . . . 5 (∅ ∈ 𝑥 ↔ ∃𝑦𝑥𝑧 ¬ 𝑧𝑦)
16 df-rex 2917 . . . . 5 (∃𝑦𝑥𝑧 ¬ 𝑧𝑦 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
1715, 16bitri 264 . . . 4 (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
18 sucel 5796 . . . . . . 7 (suc 𝑦𝑥 ↔ ∃𝑧𝑥𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))
19 df-rex 2917 . . . . . . 7 (∃𝑧𝑥𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)) ↔ ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
2018, 19bitri 264 . . . . . 6 (suc 𝑦𝑥 ↔ ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
2120imbi2i 326 . . . . 5 ((𝑦𝑥 → suc 𝑦𝑥) ↔ (𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
2221albii 1746 . . . 4 (∀𝑦(𝑦𝑥 → suc 𝑦𝑥) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
2317, 22anbi12i 733 . . 3 ((∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)) ↔ (∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))))
2423exbii 1773 . 2 (∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)) ↔ ∃𝑥(∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))))
2514, 24mpbi 220 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 196   ∨ wo 383   ∧ wa 384  ∀wal 1480   = wceq 1482  ∃wex 1703   ∈ wcel 1989  ∃wrex 2912  ∅c0 3913  suc csuc 5723  ωcom 7062 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-rep 4769  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946  ax-reg 8494  ax-inf 8532 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-om 7063  df-wrecs 7404  df-recs 7465  df-rdg 7503 This theorem is referenced by: (None)
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