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Theorem inf5 8305
 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 8297). 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 8303 . 2 ω ∈ V
2 infeq5i 8296 . 2 (ω ∈ V → ∃𝑥 𝑥 𝑥)
31, 2ax-mp 5 1 𝑥 𝑥 𝑥
 Colors of variables: wff setvar class Syntax hints:  ∃wex 1694   ∈ wcel 1938  Vcvv 3077   ⊊ wpss 3445  ∪ cuni 4270  ωcom 6838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1700  ax-4 1713  ax-5 1793  ax-6 1838  ax-7 1885  ax-8 1940  ax-9 1947  ax-10 1966  ax-11 1971  ax-12 1983  ax-13 2137  ax-ext 2494  ax-sep 4607  ax-nul 4616  ax-pr 4732  ax-un 6728  ax-inf2 8301 This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1699  df-sb 1831  df-eu 2366  df-mo 2367  df-clab 2501  df-cleq 2507  df-clel 2510  df-nfc 2644  df-ne 2686  df-ral 2805  df-rex 2806  df-rab 2809  df-v 3079  df-sbc 3307  df-dif 3447  df-un 3449  df-in 3451  df-ss 3458  df-pss 3460  df-nul 3778  df-if 3940  df-pw 4013  df-sn 4029  df-pr 4031  df-tp 4033  df-op 4035  df-uni 4271  df-br 4482  df-opab 4542  df-tr 4579  df-eprel 4843  df-po 4853  df-so 4854  df-fr 4891  df-we 4893  df-ord 5533  df-on 5534  df-lim 5535  df-suc 5536  df-om 6839 This theorem is referenced by: (None)
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