Theorem inf5 9616

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 9608). 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 9614	. 2 ⊢ ω ∈ V
2		infeq5i 9607	. 2 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 ⊊ ∪ 𝑥

Syntax hints:  ∃wex 1779   ∈ wcel 2109  Vcvv 3455   ⊊ wpss 3923  ∪ cuni 4879  ωcom 7850

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5259  ax-nul 5269  ax-pr 5395  ax-un 7718  ax-inf2 9612

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2928  df-ral 3047  df-rex 3056  df-rab 3412  df-v 3457  df-dif 3925  df-un 3927  df-in 3929  df-ss 3939  df-pss 3942  df-nul 4305  df-if 4497  df-pw 4573  df-sn 4598  df-pr 4600  df-op 4604  df-uni 4880  df-br 5116  df-opab 5178  df-tr 5223  df-eprel 5546  df-po 5554  df-so 5555  df-fr 5599  df-we 5601  df-ord 6343  df-on 6344  df-lim 6345  df-suc 6346  df-om 7851

This theorem is referenced by: (None)