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

Theorem zfcndinf 10617
Description: Axiom of Infinity ax-inf 9637, reproved from conditionless ZFC axioms. Since we have already reproved Extensionality, Replacement, and Power Sets above, we are justified in referencing Theorem el 5437 in the proof. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by NM, 15-Aug-2003.)
Assertion
Ref Expression
zfcndinf 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Distinct variable group:   𝑥,𝑦,𝑧,𝑤

Proof of Theorem zfcndinf
StepHypRef Expression
1 el 5437 . . 3 𝑤 𝑥𝑤
2 nfv 1916 . . . . . 6 𝑤 𝑥𝑦
3 nfe1 2146 . . . . . . . 8 𝑤𝑤(𝑥𝑤𝑤𝑦)
42, 3nfim 1898 . . . . . . 7 𝑤(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))
54nfal 2315 . . . . . 6 𝑤𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))
62, 5nfan 1901 . . . . 5 𝑤(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
76nfex 2316 . . . 4 𝑤𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
8 axinfnd 10605 . . . . 5 𝑦(𝑥𝑤 → (𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
9819.37iv 1951 . . . 4 (𝑥𝑤 → ∃𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
107, 9exlimi 2209 . . 3 (∃𝑤 𝑥𝑤 → ∃𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
111, 10ax-mp 5 . 2 𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
12 elequ1 2112 . . . . . 6 (𝑧 = 𝑥 → (𝑧𝑦𝑥𝑦))
13 elequ1 2112 . . . . . . . 8 (𝑧 = 𝑥 → (𝑧𝑤𝑥𝑤))
1413anbi1d 629 . . . . . . 7 (𝑧 = 𝑥 → ((𝑧𝑤𝑤𝑦) ↔ (𝑥𝑤𝑤𝑦)))
1514exbidv 1923 . . . . . 6 (𝑧 = 𝑥 → (∃𝑤(𝑧𝑤𝑤𝑦) ↔ ∃𝑤(𝑥𝑤𝑤𝑦)))
1612, 15imbi12d 344 . . . . 5 (𝑧 = 𝑥 → ((𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)) ↔ (𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
1716cbvalvw 2038 . . . 4 (∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)) ↔ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦)))
1817anbi2i 622 . . 3 ((𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))) ↔ (𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
1918exbii 1849 . 2 (∃𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦))) ↔ ∃𝑦(𝑥𝑦 ∧ ∀𝑥(𝑥𝑦 → ∃𝑤(𝑥𝑤𝑤𝑦))))
2011, 19mpbir 230 1 𝑦(𝑥𝑦 ∧ ∀𝑧(𝑧𝑦 → ∃𝑤(𝑧𝑤𝑤𝑦)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  wal 1538   = wceq 1540  wex 1780  wcel 2105
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-13 2370  ax-ext 2702  ax-sep 5299  ax-pr 5427  ax-reg 9591  ax-inf 9637
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1543  df-ex 1781  df-nf 1785  df-sb 2067  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ral 3061  df-rex 3070  df-v 3475  df-un 3953  df-sn 4629  df-pr 4631
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator