Theorem inf5 9110

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 9102). 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 9108	. 2 ⊢ ω ∈ V
2		infeq5i 9101	. 2 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 ⊊ ∪ 𝑥

Syntax hints:  ∃wex 1780   ∈ wcel 2114  Vcvv 3496   ⊊ wpss 3939  ∪ cuni 4840  ωcom 7582

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463  ax-inf2 9106

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-ord 6196  df-on 6197  df-lim 6198  df-suc 6199  df-om 7583

This theorem is referenced by: (None)