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Theorem inf5 9096
 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 9088). 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 9094 . 2 ω ∈ V
2 infeq5i 9087 . 2 (ω ∈ V → ∃𝑥 𝑥 𝑥)
31, 2ax-mp 5 1 𝑥 𝑥 𝑥
 Colors of variables: wff setvar class Syntax hints:  ∃wex 1781   ∈ wcel 2112  Vcvv 3444   ⊊ wpss 3885  ∪ cuni 4803  ωcom 7564 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298  ax-un 7445  ax-inf2 9092 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-tr 5140  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-om 7565 This theorem is referenced by: (None)
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