Theorem domen1 4472

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 4405	. . . 4 |- (B e. D -> (A ~~ B -> B ~~ A))
2	1	imp 350	. . 3 |- ((B e. D /\ A ~~ B) -> B ~~ A)
3		endomtr 4414	. . . 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 4414	. . . 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 515	1 |- ((B e. D /\ A ~~ B) -> (A ~<_ C <-> B ~<_ C))

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   e. wcel 957   class class class wbr 2616   ~~ cen 4361   ~<_ cdom 4362

This theorem is referenced by:  cdadom1 4920  iunctb 7554

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 961  ax-gen 962  ax-8 963  ax-9 964  ax-10 965  ax-11 966  ax-12 967  ax-13 968  ax-14 969  ax-17 970  ax-4 972  ax-5o 974  ax-6o 977  ax-9o 1122  ax-10o 1139  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-rep 2690  ax-sep 2700  ax-pow 2739  ax-pr 2776  ax-un 2863

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 980  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1586  df-rex 1649  df-v 1810  df-dif 2047  df-un 2048  df-in 2049  df-ss 2051  df-nul 2279  df-pw 2400  df-sn 2410  df-pr 2411  df-op 2414  df-uni 2501  df-br 2617  df-opab 2664  df-id 2832  df-xp 3181  df-rel 3182  df-cnv 3183  df-co 3184  df-dm 3185  df-rn 3186  df-res 3187  df-ima 3188  df-fun 3189  df-fn 3190  df-f 3191  df-f1 3192  df-fo 3193  df-f1o 3194  df-er 4258  df-en 4364  df-dom 4365

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