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Theorem inf5 8707
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 8699). 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 8705 . 2 ω ∈ V
2 infeq5i 8698 . 2 (ω ∈ V → ∃𝑥 𝑥 𝑥)
31, 2ax-mp 5 1 𝑥 𝑥 𝑥
Colors of variables: wff setvar class
Syntax hints:  wex 1852  wcel 2145  Vcvv 3351  wpss 3725   cuni 4575  ωcom 7213
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035  ax-un 7097  ax-inf2 8703
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-pss 3740  df-nul 4065  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-tr 4888  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-ord 5870  df-on 5871  df-lim 5872  df-suc 5873  df-om 7214
This theorem is referenced by: (None)
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