Theorem sdomdomtr 4449

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 4368	. . . 4 |- (A ~< B -> -. A ~~ B)
2	1	ad2antrl 406	. . 3 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> -. A ~~ B)
3		domtr 4396	. . . . . . . 8 |- ((A ~<_ B /\ B ~<_ C) -> A ~<_ C)
4		sdomdom 4367	. . . . . . . 8 |- (A ~< B -> A ~<_ B)
5	3, 4	sylan 448	. . . . . . 7 |- ((A ~< B /\ B ~<_ C) -> A ~<_ C)
6		brdom2 4369	. . . . . . . 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 4392	. . . . . . . . . . 11 |- (C e. D -> (A ~~ C -> C ~~ A))
12		endom 4366	. . . . . . . . . . 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 600	. . . . . . . 8 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (C ~<_ A /\ A ~<_ B)))
17		domtr 4396	. . . . . . . 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 600	. . . . . 6 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (C ~<_ B /\ B ~<_ C)))
21		sbth 4437	. . . . . 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 598	. . . 4 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (A ~~ C /\ C ~~ B)))
24		entrt 4395	. . . 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 955   class class class wbr 2609   ~~ cen 4348   ~<_ cdom 4349   ~< csdm 4350

This theorem is referenced by:  sdomentr 4450  sdomtr 4454  sucdomi 4503  infsdomnn 4511  fodomfib 4541  fodomb 4772  sucdom 4814

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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