Theorem domen1 4479

Description: Equality-like theorem for equinumerosity and dominance.

Assertion

Ref	Expression
domen1	|- ((B e. D /\ A ~~ B) -> (A ~<_ C <-> B ~<_ C))

Proof of Theorem domen1

Step	Hyp	Ref	Expression
1		ensymg 4411	. . . 4 |- (B e. D -> (A ~~ B -> B ~~ A))
2	1	imp 350	. . 3 |- ((B e. D /\ A ~~ B) -> B ~~ A)
3		endomtr 4420	. . . 4 |- ((B ~~ A /\ A ~<_ C) -> B ~<_ C)
4	3	ex 373	. . 3 |- (B ~~ A -> (A ~<_ C -> B ~<_ C))
5	2, 4	syl 10	. 2 |- ((B e. D /\ A ~~ B) -> (A ~<_ C -> B ~<_ C))
6		endomtr 4420	. . . 4 |- ((A ~~ B /\ B ~<_ C) -> A ~<_ C)
7	6	ex 373	. . 3 |- (A ~~ B -> (B ~<_ C -> A ~<_ C))
8	7	adantl 388	. 2 |- ((B e. D /\ A ~~ B) -> (B ~<_ C -> A ~<_ C))
9	5, 8	impbid 516	1 |- ((B e. D /\ A ~~ B) -> (A ~<_ C <-> B ~<_ C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 958   class class class wbr 2619   ~~ cen 4364   ~<_ cdom 4365

This theorem is referenced by:  cdadom1 4933  iunctb 7575

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-9 965  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2693  ax-sep 2703  ax-pow 2742  ax-pr 2779  ax-un 2866

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-f 3194  df-f1 3195  df-fo 3196  df-f1o 3197  df-er 4261  df-en 4368  df-dom 4369

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