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Theorem zfinf2 7589
 Description: A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 7588 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 7588 . 2
2 0el 3636 . . . . 5
3 df-rex 2703 . . . . 5
42, 3bitri 241 . . . 4
5 sucel 4646 . . . . . . 7
6 df-rex 2703 . . . . . . 7
75, 6bitri 241 . . . . . 6
87ralbii 2721 . . . . 5
9 df-ral 2702 . . . . 5
108, 9bitri 241 . . . 4
114, 10anbi12i 679 . . 3
1211exbii 1592 . 2
131, 12mpbir 201 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wo 358   wa 359  wal 1549  wex 1550   wcel 1725  wral 2697  wrex 2698  c0 3620   csuc 4575 This theorem is referenced by:  omex  7590 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-inf2 7588 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-nul 3621  df-sn 3812  df-suc 4579
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