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Theorem inf2 7534
 Description: Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using zfinf 7550 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.)
Hypothesis
Ref Expression
inf1.1
Assertion
Ref Expression
inf2
Distinct variable group:   ,,

Proof of Theorem inf2
StepHypRef Expression
1 inf1.1 . . 3
21inf1 7533 . 2
3 dfss2 3297 . . . . 5
4 eluni 3978 . . . . . . 7
54imbi2i 304 . . . . . 6
65albii 1572 . . . . 5
73, 6bitri 241 . . . 4
87anbi2i 676 . . 3
98exbii 1589 . 2
102, 9mpbir 201 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1546  wex 1547   wcel 1721   wne 2567   wss 3280  c0 3588  cuni 3975 This theorem is referenced by:  axinf2  7551 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 This theorem depends on definitions:  df-bi 178  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-v 2918  df-dif 3283  df-in 3287  df-ss 3294  df-nul 3589  df-uni 3976
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