MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  axinf2 Structured version   Visualization version   GIF version

Theorem axinf2 8700
Description: A standard version of Axiom of Infinity, expanded to primitives, derived from our version of Infinity ax-inf 8698 and Regularity ax-reg 8652.

This theorem should not be referenced in any proof. Instead, use ax-inf2 8701 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 7231 . . 3 ∅ ∈ ω
2 peano2 7232 . . . 4 (𝑦 ∈ ω → suc 𝑦 ∈ ω)
32ax-gen 1869 . . 3 𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)
4 zfinf 8699 . . . . . 6 𝑥(𝑦𝑥 ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑦𝑧𝑧𝑥)))
54inf2 8683 . . . . 5 𝑥(𝑥 ≠ ∅ ∧ 𝑥 𝑥)
65inf3 8695 . . . 4 ω ∈ V
7 eleq2 2838 . . . . 5 (𝑥 = ω → (∅ ∈ 𝑥 ↔ ∅ ∈ ω))
8 eleq2 2838 . . . . . . 7 (𝑥 = ω → (𝑦𝑥𝑦 ∈ ω))
9 eleq2 2838 . . . . . . 7 (𝑥 = ω → (suc 𝑦𝑥 ↔ suc 𝑦 ∈ ω))
108, 9imbi12d 333 . . . . . 6 (𝑥 = ω → ((𝑦𝑥 → suc 𝑦𝑥) ↔ (𝑦 ∈ ω → suc 𝑦 ∈ ω)))
1110albidv 2000 . . . . 5 (𝑥 = ω → (∀𝑦(𝑦𝑥 → suc 𝑦𝑥) ↔ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)))
127, 11anbi12d 608 . . . 4 (𝑥 = ω → ((∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)) ↔ (∅ ∈ ω ∧ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω))))
136, 12spcev 3449 . . 3 ((∅ ∈ ω ∧ ∀𝑦(𝑦 ∈ ω → suc 𝑦 ∈ ω)) → ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)))
141, 3, 13mp2an 664 . 2 𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥))
15 0el 4084 . . . . 5 (∅ ∈ 𝑥 ↔ ∃𝑦𝑥𝑧 ¬ 𝑧𝑦)
16 df-rex 3066 . . . . 5 (∃𝑦𝑥𝑧 ¬ 𝑧𝑦 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
1715, 16bitri 264 . . . 4 (∅ ∈ 𝑥 ↔ ∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦))
18 sucel 5941 . . . . . . 7 (suc 𝑦𝑥 ↔ ∃𝑧𝑥𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))
19 df-rex 3066 . . . . . . 7 (∃𝑧𝑥𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)) ↔ ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
2018, 19bitri 264 . . . . . 6 (suc 𝑦𝑥 ↔ ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))
2120imbi2i 325 . . . . 5 ((𝑦𝑥 → suc 𝑦𝑥) ↔ (𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
2221albii 1894 . . . 4 (∀𝑦(𝑦𝑥 → suc 𝑦𝑥) ↔ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦)))))
2317, 22anbi12i 604 . . 3 ((∅ ∈ 𝑥 ∧ ∀𝑦(𝑦𝑥 → suc 𝑦𝑥)) ↔ (∃𝑦(𝑦𝑥 ∧ ∀𝑧 ¬ 𝑧𝑦) ∧ ∀𝑦(𝑦𝑥 → ∃𝑧(𝑧𝑥 ∧ ∀𝑤(𝑤𝑧 ↔ (𝑤𝑦𝑤 = 𝑦))))))
2423exbii 1923 . 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  wa 382  wo 826  wal 1628   = wceq 1630  wex 1851  wcel 2144  wrex 3061  c0 4061  suc csuc 5868  ωcom 7211
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-rep 4902  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095  ax-reg 8652  ax-inf 8698
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  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 7212  df-wrecs 7558  df-recs 7620  df-rdg 7658
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator