Theorem domen1 7732

Description: Equality-like theorem for equinumerosity and dominance. (Contributed by NM, 8-Nov-2003.)

Assertion

Ref	Expression
domen1	|- ( A ~~ B -> ( A ~<_ C <-> B ~<_ C ) )

Proof of Theorem domen1

Step	Hyp	Ref	Expression
1		ensym 7636	. . 3 |- ( A ~~ B -> B ~~ A )
2		endomtr 7645	. . 3 |- ( ( B ~~ A /\ A ~<_ C ) -> B ~<_ C )
3	1, 2	sylan 479	. 2 |- ( ( A ~~ B /\ A ~<_ C ) -> B ~<_ C )
4		endomtr 7645	. 2 |- ( ( A ~~ B /\ B ~<_ C ) -> A ~<_ C )
5	3, 4	impbida 850	1 |- ( A ~~ B -> ( A ~<_ C <-> B ~<_ C ) )

Syntax hints:   -> wi 4   <-> wb 189   class class class wbr 4395   ~~ cen 7584   ~<_ cdom 7585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602

This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-er 7381  df-en 7588  df-dom 7589

This theorem is referenced by:  unxpwdom2  8121  carddomi2  8422  cdadom2  8635  cdainf  8640  cdalepw  8644  pwcdadom  8664  gchpwdom  9113  hargch  9116  dis2ndc  20552