Theorem domsdomtr 4456

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 4369	. . 3 |- (A ~<_ B <-> (A ~< B \/ A ~~ B))
2		sdomtr 4454	. . . . 5 |- ((A ~< B /\ B ~< C) -> A ~< C)
3	2	ex 373	. . . 4 |- (A ~< B -> (B ~< C -> A ~< C))
4		relsdom 4356	. . . . . . 7 |- Rel ~<
5	4	brrelexi 3198	. . . . . 6 |- (B ~< C -> B e. V)
6		endomtr 4401	. . . . . . . . . . 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 4392	. . . . . . . . . . . 12 |- (B e. V -> (A ~~ B -> B ~~ A))
10		entrt 4395	. . . . . . . . . . . . 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 556	. . . . . . . 8 |- ((B e. V /\ A ~~ B) -> ((B ~<_ C /\ -. B ~~ C) -> (A ~<_ C /\ -. A ~~ C)))
16		brsdom 4363	. . . . . . . 8 |- (B ~< C <-> (B ~<_ C /\ -. B ~~ C))
17		brsdom 4363	. . . . . . . 8 |- (A ~< C <-> (A ~<_ C /\ -. A ~~ C))
18	15, 16, 17	3imtr4g 551	. . . . . . 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 955  Vcvv 1802   class class class wbr 2609   ~~ cen 4348   ~<_ cdom 4349   ~< csdm 4350

This theorem is referenced by:  2pwuninel 4465  pwuninelg 4467  ondomon 4828  ondomcard 4829  cardmin 4832  alephsucdom 4852  infdif 7511

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-rep 2683  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fn 3183  df-f 3184  df-f1 3185  df-fo 3186  df-f1o 3187  df-er 4245  df-en 4351  df-dom 4352  df-sdom 4353

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