Theorem domsdomtr 4482

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 4394	. . 3 |- (A ~<_ B <-> (A ~< B \/ A ~~ B))
2		sdomtr 4480	. . . . 5 |- ((A ~< B /\ B ~< C) -> A ~< C)
3	2	ex 373	. . . 4 |- (A ~< B -> (B ~< C -> A ~< C))
4		relsdom 4380	. . . . . . 7 |- Rel ~<
5	4	brrelexi 3214	. . . . . 6 |- (B ~< C -> B e. V)
6		endomtr 4426	. . . . . . . . . . 11 |- ((A ~~ B /\ B ~<_ C) -> A ~<_ C)
7	6	ex 373	. . . . . . . . . 10 |- (A ~~ B -> (B ~<_ C -> A ~<_ C))
8	7	adantl 390	. . . . . . . . 9 |- ((B e. V /\ A ~~ B) -> (B ~<_ C -> A ~<_ C))
9		ensymg 4417	. . . . . . . . . . . 12 |- (B e. V -> (A ~~ B -> B ~~ A))
10		entrt 4420	. . . . . . . . . . . . 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 560	. . . . . . . 8 |- ((B e. V /\ A ~~ B) -> ((B ~<_ C /\ -. B ~~ C) -> (A ~<_ C /\ -. A ~~ C)))
16		brsdom 4387	. . . . . . . 8 |- (B ~< C <-> (B ~<_ C /\ -. B ~~ C))
17		brsdom 4387	. . . . . . . 8 |- (A ~< C <-> (A ~<_ C /\ -. A ~~ C))
18	15, 16, 17	3imtr4g 555	. . . . . . 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 960  Vcvv 1814   class class class wbr 2624   ~~ cen 4370   ~<_ cdom 4371   ~< csdm 4372

This theorem is referenced by:  pwuninel 4492  2pwuninel 4493  ondomon 4867  ondomcard 4868  cardmin 4871  alephsucdom 4891  infdif 7569

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-er 4267  df-en 4374  df-dom 4375  df-sdom 4376

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