MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  zfinf2 Unicode version

Theorem zfinf2 8080
Description: A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 8079 for the unabbreviated version.) (Contributed by NM, 30-Aug-1993.)
Assertion
Ref Expression
zfinf2
Distinct variable group:   ,

Proof of Theorem zfinf2
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ax-inf2 8079 . 2
2 0el 3802 . . . . 5
3 df-rex 2813 . . . . 5
42, 3bitri 249 . . . 4
5 sucel 4956 . . . . . . 7
6 df-rex 2813 . . . . . . 7
75, 6bitri 249 . . . . . 6
87ralbii 2888 . . . . 5
9 df-ral 2812 . . . . 5
108, 9bitri 249 . . . 4
114, 10anbi12i 697 . . 3
1211exbii 1667 . 2
131, 12mpbir 209 1
Colors of variables: wff setvar class
Syntax hints:  -.wn 3  ->wi 4  <->wb 184  \/wo 368  /\wa 369  A.wal 1393  E.wex 1612  e.wcel 1818  A.wral 2807  E.wrex 2808   c0 3784  succsuc 4885
This theorem is referenced by:  omex  8081
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-inf2 8079
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-v 3111  df-dif 3478  df-un 3480  df-nul 3785  df-sn 4030  df-suc 4889
  Copyright terms: Public domain W3C validator