Theorem domen2 7004

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

Assertion

Ref	Expression
domen2	|- ( A ~~ B -> ( C ~<_ A <-> C ~<_ B ) )

Proof of Theorem domen2

Step	Hyp	Ref	Expression
1		domentr 6943	. . 3 |- ( ( C ~<_ A /\ A ~~ B ) -> C ~<_ B )
2	1	ancoms 268	. 2 |- ( ( A ~~ B /\ C ~<_ A ) -> C ~<_ B )
3		ensym 6933	. . 3 |- ( A ~~ B -> B ~~ A )
4		domentr 6943	. . . 4 |- ( ( C ~<_ B /\ B ~~ A ) -> C ~<_ A )
5	4	ancoms 268	. . 3 |- ( ( B ~~ A /\ C ~<_ B ) -> C ~<_ A )
6	3, 5	sylan 283	. 2 |- ( ( A ~~ B /\ C ~<_ B ) -> C ~<_ A )
7	2, 6	impbida 598	1 |- ( A ~~ B -> ( C ~<_ A <-> C ~<_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   class class class wbr 4083   ~~ cen 6885   ~<_ cdom 6886

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-er 6680  df-en 6888  df-dom 6889

This theorem is referenced by:  fihashdom  11025