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Theorem inf5 9670
Description: The statement "there exists a set that is a proper subset of its union" is equivalent to the Axiom of Infinity (see Theorem infeq5 9662). This provides us with a very compact way to express the Axiom of Infinity using only elementary symbols. (Contributed by NM, 3-Jun-2005.)
Assertion
Ref Expression
inf5 𝑥 𝑥 𝑥

Proof of Theorem inf5
StepHypRef Expression
1 omex 9668 . 2 ω ∈ V
2 infeq5i 9661 . 2 (ω ∈ V → ∃𝑥 𝑥 𝑥)
31, 2ax-mp 5 1 𝑥 𝑥 𝑥
Colors of variables: wff setvar class
Syntax hints:  wex 1773  wcel 2098  Vcvv 3461  wpss 3945   cuni 4909  ωcom 7871
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696  ax-sep 5300  ax-nul 5307  ax-pr 5429  ax-un 7741  ax-inf2 9666
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-ral 3051  df-rex 3060  df-rab 3419  df-v 3463  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3964  df-nul 4323  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4910  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5582  df-po 5590  df-so 5591  df-fr 5633  df-we 5635  df-ord 6374  df-on 6375  df-lim 6376  df-suc 6377  df-om 7872
This theorem is referenced by: (None)
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