Theorem sdomdomtr 3370

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 3291	. . . 4 |- (A ~< B -> -. A ~~ B)
2	1	ad2antrl 322	. . 3 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> -. A ~~ B)
3		domtr 3320	. . . . . . . 8 |- ((A ~<_ B /\ B ~<_ C) -> A ~<_ C)
4		sdomdom 3290	. . . . . . . 8 |- (A ~< B -> A ~<_ B)
5	3, 4	sylan 343	. . . . . . 7 |- ((A ~< B /\ B ~<_ C) -> A ~<_ C)
6		brdom2 3292	. . . . . . . 8 |- (A ~<_ C <-> (A ~< C \/ A ~~ C))
7		df-or 197	. . . . . . . 8 |- ((A ~< C \/ A ~~ C) <-> (-. A ~< C -> A ~~ C))
8	6, 7	bitr 151	. . . . . . 7 |- (A ~<_ C <-> (-. A ~< C -> A ~~ C))
9	5, 8	sylib 173	. . . . . 6 |- ((A ~< B /\ B ~<_ C) -> (-. A ~< C -> A ~~ C))
10	9	adantl 305	. . . . 5 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> A ~~ C))
11		ensymg 3316	. . . . . . . . . . 11 |- (C e. D -> (A ~~ C -> C ~~ A))
12		endom 3289	. . . . . . . . . . 11 |- (C ~~ A -> C ~<_ A)
13	11, 12	syl6 23	. . . . . . . . . 10 |- (C e. D -> (A ~~ C -> C ~<_ A))
14	9, 13	sylan9r 360	. . . . . . . . 9 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> C ~<_ A))
15	4	ad2antrl 322	. . . . . . . . . 10 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> A ~<_ B)
16	15	a1d 14	. . . . . . . . 9 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> A ~<_ B))
17	14, 16	jcad 455	. . . . . . . 8 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (C ~<_ A /\ A ~<_ B)))
18		domtr 3320	. . . . . . . 8 |- ((C ~<_ A /\ A ~<_ B) -> C ~<_ B)
19	17, 18	syl6 23	. . . . . . 7 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> C ~<_ B))
20		pm3.27 260	. . . . . . . . 9 |- ((A ~< B /\ B ~<_ C) -> B ~<_ C)
21	20	adantl 305	. . . . . . . 8 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> B ~<_ C)
22	21	a1d 14	. . . . . . 7 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> B ~<_ C))
23	19, 22	jcad 455	. . . . . 6 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (C ~<_ B /\ B ~<_ C)))
24		sbth 3359	. . . . . 6 |- ((C ~<_ B /\ B ~<_ C) -> C ~~ B)
25	23, 24	syl6 23	. . . . 5 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> C ~~ B))
26	10, 25	jcad 455	. . . 4 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> (A ~~ C /\ C ~~ B)))
27		entrt 3319	. . . 4 |- ((A ~~ C /\ C ~~ B) -> A ~~ B)
28	26, 27	syl6 23	. . 3 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> (-. A ~< C -> A ~~ B))
29	2, 28	mt3d 101	. 2 |- ((C e. D /\ (A ~< B /\ B ~<_ C)) -> A ~< C)
30	29	exp 291	1 |- (C e. D -> ((A ~< B /\ B ~<_ C) -> A ~< C))

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195   /\ wa 196   e. wcel 1092   class class class wbr 2054   ~~ cen 3271   ~<_ cdom 3272   ~< csdm 3273

This theorem is referenced by:  sdomentr 3371  sdomtr 3373  sucdomi 3419  infsdomnn 3426  fodomb 3615  sucdom 3648

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-er 3200  df-en 3274  df-dom 3275  df-sdom 3276

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