Theorem sdomdomtr 4455

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 4374	. . . 4 |- (A ~< B -> -. A ~~ B)
2	1	ad2antrl 406	. . 3 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> -. A ~~ B)
3		domtr 4402	. . . . . . . 8 |- ((A ~<_ B /\ B ~<_ C) -> A ~<_ C)
4		sdomdom 4373	. . . . . . . 8 |- (A ~< B -> A ~<_ B)
5	3, 4	sylan 448	. . . . . . 7 |- ((A ~< B /\ B ~<_ C) -> A ~<_ C)
6		brdom2 4375	. . . . . . . 8 |- (A ~<_ C <-> (A ~< C \/ A ~~ C))
7		df-or 224	. . . . . . . 8 |- ((A ~< C \/ A ~~ C) <-> (-. A ~< C -> A ~~ C))
8	6, 7	bitr 173	. . . . . . 7 |- (A ~<_ C <-> (-. A ~< C -> A ~~ C))
9	5, 8	sylib 198	. . . . . 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 4398	. . . . . . . . . . 11 |- (C e. D -> (A ~~ C -> C ~~ A))
12		endom 4372	. . . . . . . . . . 11 |- (C ~~ A -> C ~<_ A)
13	11, 12	syl6 22	. . . . . . . . . 10 |- (C e. D -> (A ~~ C -> C ~<_ A))
14	9, 13	sylan9r 469	. . . . . . . . 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 601	. . . . . . . 8 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (C ~<_ A /\ A ~<_ B)))
17		domtr 4402	. . . . . . . 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 601	. . . . . 6 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (C ~<_ B /\ B ~<_ C)))
21		sbth 4443	. . . . . 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 599	. . . 4 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (A ~~ C /\ C ~~ B)))
24		entrt 4401	. . . 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 114	. 2 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> A ~< C)
27	26	ex 373	1 |- (C e. D -> ((A ~< B /\ B ~<_ C) -> A ~< C))

Syntax hints:  -. wn 2   -> wi 3   \/ wo 222   /\ wa 223   e. wcel 956   class class class wbr 2614   ~~ cen 4354   ~<_ cdom 4355   ~< csdm 4356

This theorem is referenced by:  sdomentr 4456  sdomtr 4460  sucdomi 4509  infsdomnn 4517  fodomfib 4547  fodomb 4780  sucdom 4822

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2688  ax-sep 2698  ax-pow 2737  ax-pr 2774  ax-un 2861

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-id 2830  df-xp 3179  df-rel 3180  df-cnv 3181  df-co 3182  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fun 3187  df-fn 3188  df-f 3189  df-f1 3190  df-fo 3191  df-f1o 3192  df-er 4251  df-en 4357  df-dom 4358  df-sdom 4359

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