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Theorem zfinf2 7553
 Description: A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (See ax-inf2 7552 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 7552 . 2
2 0el 3604 . . . . 5
3 df-rex 2672 . . . . 5
42, 3bitri 241 . . . 4
5 sucel 4614 . . . . . . 7
6 df-rex 2672 . . . . . . 7
75, 6bitri 241 . . . . . 6
87ralbii 2690 . . . . 5
9 df-ral 2671 . . . . 5
108, 9bitri 241 . . . 4
114, 10anbi12i 679 . . 3
1211exbii 1589 . 2
131, 12mpbir 201 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wo 358   wa 359  wal 1546  wex 1547   wcel 1721  wral 2666  wrex 2667  c0 3588   csuc 4543 This theorem is referenced by:  omex  7554 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-inf2 7552 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-v 2918  df-dif 3283  df-un 3285  df-nul 3589  df-sn 3780  df-suc 4547
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