Theorem domen2 7098

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 7033	. . 3 |- ( ( C ~<_ A /\ A ~~ B ) -> C ~<_ B )
2	1	ancoms 268	. 2 |- ( ( A ~~ B /\ C ~<_ A ) -> C ~<_ B )
3		ensym 7023	. . 3 |- ( A ~~ B -> B ~~ A )
4		domentr 7033	. . . 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 600	1 |- ( A ~~ B -> ( C ~<_ A <-> C ~<_ B ) )

Syntax hints:   -> wi 4   <-> wb 105   class class class wbr 4111   ~~ cen 6975   ~<_ cdom 6976

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-id 4416  df-xp 4757  df-rel 4758  df-cnv 4759  df-co 4760  df-dm 4761  df-rn 4762  df-res 4763  df-ima 4764  df-fun 5356  df-fn 5357  df-f 5358  df-f1 5359  df-fo 5360  df-f1o 5361  df-er 6769  df-en 6978  df-dom 6979

This theorem is referenced by:  fihashdom  11171