Theorem domsdomtr 3374

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

Assertion

Ref	Expression
domsdomtr	|- ((A ~<_ B /\ B ~< C) -> A ~< C)

Proof of Theorem domsdomtr

Step	Hyp	Ref	Expression
1		brdom2 3292	. . 3 |- (A ~<_ B <-> (A ~< B \/ A ~~ B))
2		sdomtr 3373	. . . . 5 |- ((A ~< B /\ B ~< C) -> A ~< C)
3	2	exp 291	. . . 4 |- (A ~< B -> (B ~< C -> A ~< C))
4		relsdom 3279	. . . . . . 7 |- Rel ~<
5	4	brrelexi 2447	. . . . . 6 |- (B ~< C -> B e. V)
6		endomtr 3325	. . . . . . . . . . 11 |- ((A ~~ B /\ B ~<_ C) -> A ~<_ C)
7	6	exp 291	. . . . . . . . . 10 |- (A ~~ B -> (B ~<_ C -> A ~<_ C))
8	7	adantl 305	. . . . . . . . 9 |- ((B e. V /\ A ~~ B) -> (B ~<_ C -> A ~<_ C))
9		ensymg 3316	. . . . . . . . . . . 12 |- (B e. V -> (A ~~ B -> B ~~ A))
10		entrt 3319	. . . . . . . . . . . . 13 |- ((B ~~ A /\ A ~~ C) -> B ~~ C)
11	10	exp 291	. . . . . . . . . . . 12 |- (B ~~ A -> (A ~~ C -> B ~~ C))
12	9, 11	syl6 23	. . . . . . . . . . 11 |- (B e. V -> (A ~~ B -> (A ~~ C -> B ~~ C)))
13	12	imp 277	. . . . . . . . . 10 |- ((B e. V /\ A ~~ B) -> (A ~~ C -> B ~~ C))
14	13	con3d 87	. . . . . . . . 9 |- ((B e. V /\ A ~~ B) -> (-. B ~~ C -> -. A ~~ C))
15	8, 14	anim12d 431	. . . . . . . 8 |- ((B e. V /\ A ~~ B) -> ((B ~<_ C /\ -. B ~~ C) -> (A ~<_ C /\ -. A ~~ C)))
16		brsdom 3286	. . . . . . . 8 |- (B ~< C <-> (B ~<_ C /\ -. B ~~ C))
17		brsdom 3286	. . . . . . . 8 |- (A ~< C <-> (A ~<_ C /\ -. A ~~ C))
18	15, 16, 17	3imtr4g 426	. . . . . . 7 |- ((B e. V /\ A ~~ B) -> (B ~< C -> A ~< C))
19	18	exp 291	. . . . . 6 |- (B e. V -> (A ~~ B -> (B ~< C -> A ~< C)))
20	5, 19	syl 12	. . . . 5 |- (B ~< C -> (A ~~ B -> (B ~< C -> A ~< C)))
21	20	pm2.43b 61	. . . 4 |- (A ~~ B -> (B ~< C -> A ~< C))
22	3, 21	jaoi 275	. . 3 |- ((A ~< B \/ A ~~ B) -> (B ~< C -> A ~< C))
23	1, 22	sylbi 174	. 2 |- (A ~<_ B -> (B ~< C -> A ~< C))
24	23	imp 277	1 |- ((A ~<_ B /\ B ~< C) -> A ~< C)

Syntax hints:  -. wn 1   -> wi 2   \/ wo 195   /\ wa 196   e. wcel 1092  Vcvv 1348   class class class wbr 2054   ~~ cen 3271   ~<_ cdom 3272   ~< csdm 3273

This theorem is referenced by:  ondomon 3662  ondomcard 3663  cardmin 3666  alephsucdom 3685  infdif 4948

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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