Theorem domsdomtr 4476

Description: Transitivity of dominance and strict dominance. Theorem 22(ii) of [Suppes] p. 97.

Assertion

Ref	Expression
domsdomtr	|- ((A ~<_ B /\ B ~< C) -> A ~< C)

Proof of Theorem domsdomtr

Step	Hyp	Ref	Expression
1		brdom2 4388	. . 3 |- (A ~<_ B <-> (A ~< B \/ A ~~ B))
2		sdomtr 4474	. . . . 5 |- ((A ~< B /\ B ~< C) -> A ~< C)
3	2	ex 373	. . . 4 |- (A ~< B -> (B ~< C -> A ~< C))
4		relsdom 4374	. . . . . . 7 |- Rel ~<
5	4	brrelexi 3208	. . . . . 6 |- (B ~< C -> B e. V)
6		endomtr 4420	. . . . . . . . . . 11 |- ((A ~~ B /\ B ~<_ C) -> A ~<_ C)
7	6	ex 373	. . . . . . . . . 10 |- (A ~~ B -> (B ~<_ C -> A ~<_ C))
8	7	adantl 388	. . . . . . . . 9 |- ((B e. V /\ A ~~ B) -> (B ~<_ C -> A ~<_ C))
9		ensymg 4411	. . . . . . . . . . . 12 |- (B e. V -> (A ~~ B -> B ~~ A))
10		entrt 4414	. . . . . . . . . . . . 13 |- ((B ~~ A /\ A ~~ C) -> B ~~ C)
11	10	ex 373	. . . . . . . . . . . 12 |- (B ~~ A -> (A ~~ C -> B ~~ C))
12	9, 11	syl6 22	. . . . . . . . . . 11 |- (B e. V -> (A ~~ B -> (A ~~ C -> B ~~ C)))
13	12	imp 350	. . . . . . . . . 10 |- ((B e. V /\ A ~~ B) -> (A ~~ C -> B ~~ C))
14	13	con3d 95	. . . . . . . . 9 |- ((B e. V /\ A ~~ B) -> (-. B ~~ C -> -. A ~~ C))
15	8, 14	anim12d 558	. . . . . . . 8 |- ((B e. V /\ A ~~ B) -> ((B ~<_ C /\ -. B ~~ C) -> (A ~<_ C /\ -. A ~~ C)))
16		brsdom 4381	. . . . . . . 8 |- (B ~< C <-> (B ~<_ C /\ -. B ~~ C))
17		brsdom 4381	. . . . . . . 8 |- (A ~< C <-> (A ~<_ C /\ -. A ~~ C))
18	15, 16, 17	3imtr4g 553	. . . . . . 7 |- ((B e. V /\ A ~~ B) -> (B ~< C -> A ~< C))
19	18	ex 373	. . . . . 6 |- (B e. V -> (A ~~ B -> (B ~< C -> A ~< C)))
20	5, 19	syl 10	. . . . 5 |- (B ~< C -> (A ~~ B -> (B ~< C -> A ~< C)))
21	20	pm2.43b 67	. . . 4 |- (A ~~ B -> (B ~< C -> A ~< C))
22	3, 21	jaoi 341	. . 3 |- ((A ~< B \/ A ~~ B) -> (B ~< C -> A ~< C))
23	1, 22	sylbi 199	. 2 |- (A ~<_ B -> (B ~< C -> A ~< C))
24	23	imp 350	1 |- ((A ~<_ B /\ B ~< C) -> A ~< C)

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   e. wcel 958  Vcvv 1811   class class class wbr 2619   ~~ cen 4364   ~<_ cdom 4365   ~< csdm 4366

This theorem is referenced by:  pwuninel 4486  2pwuninel 4487  ondomon 4856  ondomcard 4857  cardmin 4860  alephsucdom 4880  infdif 7568

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-ral 1649  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  df-sdom 4370

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