Theorem sdomdomtr 5532

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 5446	. . . 4 |- (A ~< B -> -. A ~~ B)
2	1	ad2antrl 442	. . 3 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> -. A ~~ B)
3		domtr 5474	. . . . . . . 8 |- ((A ~<_ B /\ B ~<_ C) -> A ~<_ C)
4		sdomdom 5445	. . . . . . . 8 |- (A ~< B -> A ~<_ B)
5	3, 4	sylan 497	. . . . . . 7 |- ((A ~< B /\ B ~<_ C) -> A ~<_ C)
6		brdom2 5447	. . . . . . . 8 |- (A ~<_ C <-> (A ~< C \/ A ~~ C))
7		df-or 241	. . . . . . . 8 |- ((A ~< C \/ A ~~ C) <-> (-. A ~< C -> A ~~ C))
8	6, 7	bitri 190	. . . . . . 7 |- (A ~<_ C <-> (-. A ~< C -> A ~~ C))
9	5, 8	sylib 215	. . . . . 6 |- ((A ~< B /\ B ~<_ C) -> (-. A ~< C -> A ~~ C))
10	9	adantl 424	. . . . 5 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> A ~~ C))
11		ensymg 5470	. . . . . . . . . . 11 |- (C e. D -> (A ~~ C -> C ~~ A))
12		endom 5444	. . . . . . . . . . 11 |- (C ~~ A -> C ~<_ A)
13	11, 12	syl6 25	. . . . . . . . . 10 |- (C e. D -> (A ~~ C -> C ~<_ A))
14	9, 13	sylan9r 519	. . . . . . . . 9 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> C ~<_ A))
15	4	ad2antrl 442	. . . . . . . . 9 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> A ~<_ B)
16	14, 15	jctird 663	. . . . . . . 8 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (C ~<_ A /\ A ~<_ B)))
17		domtr 5474	. . . . . . . 8 |- ((C ~<_ A /\ A ~<_ B) -> C ~<_ B)
18	16, 17	syl6 25	. . . . . . 7 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> C ~<_ B))
19		simprr 451	. . . . . . 7 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> B ~<_ C)
20	18, 19	jctird 663	. . . . . 6 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (C ~<_ B /\ B ~<_ C)))
21		sbth 5520	. . . . . 6 |- ((C ~<_ B /\ B ~<_ C) -> C ~~ B)
22	20, 21	syl6 25	. . . . 5 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> C ~~ B))
23	10, 22	jcad 661	. . . 4 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (A ~~ C /\ C ~~ B)))
24		entr 5473	. . . 4 |- ((A ~~ C /\ C ~~ B) -> A ~~ B)
25	23, 24	syl6 25	. . 3 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> A ~~ B))
26	2, 25	mt3d 129	. 2 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> A ~< C)
27	26	ex 402	1 |- (C e. D -> ((A ~< B /\ B ~<_ C) -> A ~< C))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   e. wcel 1300   class class class wbr 3338   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425

This theorem is referenced by:  sdomentr 5533  sdomtr 5537  sucdomi 5617  infsdomnn 5625  fodomfib 5657  omsubsuc2 5878  omsubsdomlem2 5880  elomsubsd 5885  fodomb 5962  sucdom 5994  tarax3d2 15225  cptarc 15242  tarsuc2 15245  omsubsuc2OLD 15387  omsubsdomlem2OLD 15389  elomsubsdOLD 15394  ufilen 15579

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-13 1311  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-rep 3428  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524  ax-un 3790

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-rab 2112  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-f 4010  df-f1 4011  df-fo 4012  df-f1o 4013  df-er 5318  df-en 5427  df-dom 5428  df-sdom 5429

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