Theorem fin2inf 8223

Description: This (useless) theorem, which was proved without the Axiom of Infinity, demonstrates an artifact of our definition of binary relation, which is meaningful only when its arguments exist. In particular, the antecedent cannot be satisfied unless om exists. (Contributed by NM, 13-Nov-2003.)

Assertion

Ref	Expression
fin2inf	|- ( A ~< om -> om e. _V )

Proof of Theorem fin2inf

Step	Hyp	Ref	Expression
1		relsdom 7962	. 2 |- Rel ~<
2	1	brrelex2i 5159	1 |- ( A ~< om -> om e. _V )

Syntax hints:   -> wi 4   e. wcel 1990   _Vcvv 3200   class class class wbr 4653   omcom 7065   ~< csdm 7954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-dom 7957  df-sdom 7958

This theorem is referenced by:  unfi2  8229  unifi2  8256