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Theorem inf2 7570
 Description: Variation of Axiom of Infinity. There exists a non-empty set that is a subset of its union (using zfinf 7586 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 7569 . 2
3 dfss2 3329 . . . . 5
4 eluni 4010 . . . . . . 7
54imbi2i 304 . . . . . 6
65albii 1575 . . . . 5
73, 6bitri 241 . . . 4
87anbi2i 676 . . 3
98exbii 1592 . 2
102, 9mpbir 201 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359  wal 1549  wex 1550   wcel 1725   wne 2598   wss 3312  c0 3620  cuni 4007 This theorem is referenced by:  axinf2  7587 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 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-v 2950  df-dif 3315  df-in 3319  df-ss 3326  df-nul 3621  df-uni 4008
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