Theorem sdomdomtr 4614

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

Assertion

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

Proof of Theorem sdomdomtr

Step	Hyp	Ref	Expression
1		sdomnen 4528	. . . 4 |- (A ~< B -> -. A ~~ B)
2	1	ad2antrl 406	. . 3 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> -. A ~~ B)
3		domtr 4556	. . . . . . . 8 |- ((A ~<_ B /\ B ~<_ C) -> A ~<_ C)
4		sdomdom 4527	. . . . . . . 8 |- (A ~< B -> A ~<_ B)
5	3, 4	sylan 450	. . . . . . 7 |- ((A ~< B /\ B ~<_ C) -> A ~<_ C)
6		brdom2 4529	. . . . . . . 8 |- (A ~<_ C <-> (A ~< C \/ A ~~ C))
7		df-or 222	. . . . . . . 8 |- ((A ~< C \/ A ~~ C) <-> (-. A ~< C -> A ~~ C))
8	6, 7	bitri 171	. . . . . . 7 |- (A ~<_ C <-> (-. A ~< C -> A ~~ C))
9	5, 8	sylib 196	. . . . . 6 |- ((A ~< B /\ B ~<_ C) -> (-. A ~< C -> A ~~ C))
10	9	adantl 388	. . . . 5 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> A ~~ C))
11		ensymg 4552	. . . . . . . . . . 11 |- (C e. D -> (A ~~ C -> C ~~ A))
12		endom 4526	. . . . . . . . . . 11 |- (C ~~ A -> C ~<_ A)
13	11, 12	syl6 22	. . . . . . . . . 10 |- (C e. D -> (A ~~ C -> C ~<_ A))
14	9, 13	sylan9r 471	. . . . . . . . 9 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> C ~<_ A))
15	4	ad2antrl 406	. . . . . . . . 9 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> A ~<_ B)
16	14, 15	jctird 605	. . . . . . . 8 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (C ~<_ A /\ A ~<_ B)))
17		domtr 4556	. . . . . . . 8 |- ((C ~<_ A /\ A ~<_ B) -> C ~<_ B)
18	16, 17	syl6 22	. . . . . . 7 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> C ~<_ B))
19		simprr 415	. . . . . . 7 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> B ~<_ C)
20	18, 19	jctird 605	. . . . . 6 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (C ~<_ B /\ B ~<_ C)))
21		sbth 4602	. . . . . 6 |- ((C ~<_ B /\ B ~<_ C) -> C ~~ B)
22	20, 21	syl6 22	. . . . 5 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> C ~~ B))
23	10, 22	jcad 603	. . . 4 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (A ~~ C /\ C ~~ B)))
24		entr 4555	. . . 4 |- ((A ~~ C /\ C ~~ B) -> A ~~ B)
25	23, 24	syl6 22	. . 3 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> A ~~ B))
26	2, 25	mt3d 113	. 2 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> A ~< C)
27	26	ex 371	1 |- (C e. D -> ((A ~< B /\ B ~<_ C) -> A ~< C))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 220   /\ wa 221   e. wcel 994   class class class wbr 2692   ~~ cen 4505   ~<_ cdom 4506   ~< csdm 4507

This theorem is referenced by:  sdomentr 4615  sdomtr 4619  sucdomi 4670  infsdomnn 4678  fodomfib 4710  fodomb 4946  sucdom 4992  omsubsuc2 11439  omsubsdomlem2 11441  elomsubsd 11446  ufilen 11664

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-9 1001  ax-10 1002  ax-11 1003  ax-12 1004  ax-13 1005  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-rep 2767  ax-sep 2777  ax-pow 2818  ax-pr 2855  ax-un 3089

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 783  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-ral 1695  df-rex 1696  df-v 1858  df-dif 2101  df-un 2102  df-in 2103  df-ss 2105  df-nul 2333  df-pw 2459  df-sn 2470  df-pr 2471  df-op 2474  df-uni 2570  df-br 2693  df-opab 2741  df-id 2913  df-xp 3265  df-rel 3266  df-cnv 3267  df-co 3268  df-dm 3269  df-rn 3270  df-res 3271  df-ima 3272  df-fun 3273  df-fn 3274  df-f 3275  df-f1 3276  df-fo 3277  df-f1o 3278  df-er 4401  df-en 4509  df-dom 4510  df-sdom 4511

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