Theorem inf5 8235

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 8227). 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 8233	. 2 ⊢ ω ∈ V
2		infeq5i 8226	. 2 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥)
3	1, 2	ax-mp 5	1 ⊢ ∃𝑥 𝑥 ⊊ ∪ 𝑥

Syntax hints:  ∃wex 1692   ∈ wcel 1937  Vcvv 3066   ⊊ wpss 3427  ∪ cuni 4228  ωcom 6769

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1698  ax-4 1711  ax-5 1789  ax-6 1836  ax-7 1883  ax-8 1939  ax-9 1946  ax-10 1965  ax-11 1970  ax-12 1983  ax-13 2137  ax-ext 2485  ax-sep 4558  ax-nul 4567  ax-pr 4680  ax-un 6659  ax-inf2 8231

This theorem depends on definitions:  df-bi 192  df-or 379  df-an 380  df-3or 1022  df-3an 1023  df-tru 1471  df-ex 1693  df-nf 1697  df-sb 1829  df-eu 2357  df-mo 2358  df-clab 2492  df-cleq 2498  df-clel 2501  df-nfc 2635  df-ne 2677  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3068  df-sbc 3292  df-dif 3429  df-un 3431  df-in 3433  df-ss 3440  df-pss 3442  df-nul 3758  df-if 3909  df-pw 3980  df-sn 3996  df-pr 3998  df-tp 4000  df-op 4002  df-uni 4229  df-br 4435  df-opab 4494  df-tr 4531  df-eprel 4791  df-po 4801  df-so 4802  df-fr 4839  df-we 4841  df-ord 5477  df-on 5478  df-lim 5479  df-suc 5480  df-om 6770

This theorem is referenced by: (None)