Theorem inf5 9662

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 9654). 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 9660	. 2 ⊢ ω ∈ V
2		infeq5i 9653	. 2 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 ⊊ ∪ 𝑥

Syntax hints:  ∃wex 1774   ∈ wcel 2099  Vcvv 3470   ⊊ wpss 3946  ∪ cuni 4903  ωcom 7864

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734  ax-inf2 9658

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-om 7865

This theorem is referenced by: (None)