Theorem inf5 9566

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 9558). 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

Step	Hyp	Ref	Expression
1		omex 9564	. 2 ⊢ ω ∈ V
2		infeq5i 9557	. 2 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 ⊊ ∪ 𝑥

Syntax hints:  ∃wex 1781   ∈ wcel 2114  Vcvv 3442   ⊊ wpss 3904  ∪ cuni 4865  ωcom 7818

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690  ax-inf2 9562

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-tr 5208  df-eprel 5532  df-po 5540  df-so 5541  df-fr 5585  df-we 5587  df-ord 6328  df-on 6329  df-lim 6330  df-suc 6331  df-om 7819

This theorem is referenced by: (None)