Theorem domsdomtr 5539

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 5447	. . 3 |- (A ~<_ B <-> (A ~< B \/ A ~~ B))
2		sdomtr 5537	. . . . 5 |- ((A ~< B /\ B ~< C) -> A ~< C)
3	2	ex 402	. . . 4 |- (A ~< B -> (B ~< C -> A ~< C))
4		relsdom 5433	. . . . . . 7 |- Rel ~<
5	4	brrelexi 4029	. . . . . 6 |- (B ~< C -> B e. _V)
6		endomtr 5479	. . . . . . . . . . 11 |- ((A ~~ B /\ B ~<_ C) -> A ~<_ C)
7	6	ex 402	. . . . . . . . . 10 |- (A ~~ B -> (B ~<_ C -> A ~<_ C))
8	7	adantl 424	. . . . . . . . 9 |- ((B e. _V /\ A ~~ B) -> (B ~<_ C -> A ~<_ C))
9		ensymg 5470	. . . . . . . . . . . 12 |- (B e. _V -> (A ~~ B -> B ~~ A))
10		entr 5473	. . . . . . . . . . . . 13 |- ((B ~~ A /\ A ~~ C) -> B ~~ C)
11	10	ex 402	. . . . . . . . . . . 12 |- (B ~~ A -> (A ~~ C -> B ~~ C))
12	9, 11	syl6 25	. . . . . . . . . . 11 |- (B e. _V -> (A ~~ B -> (A ~~ C -> B ~~ C)))
13	12	imp 377	. . . . . . . . . 10 |- ((B e. _V /\ A ~~ B) -> (A ~~ C -> B ~~ C))
14	13	con3d 111	. . . . . . . . 9 |- ((B e. _V /\ A ~~ B) -> (-. B ~~ C -> -. A ~~ C))
15	8, 14	anim12d 617	. . . . . . . 8 |- ((B e. _V /\ A ~~ B) -> ((B ~<_ C /\ -. B ~~ C) -> (A ~<_ C /\ -. A ~~ C)))
16		brsdom 5440	. . . . . . . 8 |- (B ~< C <-> (B ~<_ C /\ -. B ~~ C))
17		brsdom 5440	. . . . . . . 8 |- (A ~< C <-> (A ~<_ C /\ -. A ~~ C))
18	15, 16, 17	3imtr4g 612	. . . . . . 7 |- ((B e. _V /\ A ~~ B) -> (B ~< C -> A ~< C))
19	18	ex 402	. . . . . 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 81	. . . 4 |- (A ~~ B -> (B ~< C -> A ~< C))
22	3, 21	jaoi 368	. . 3 |- ((A ~< B \/ A ~~ B) -> (B ~< C -> A ~< C))
23	1, 22	sylbi 216	. 2 |- (A ~<_ B -> (B ~< C -> A ~< C))
24	23	imp 377	1 |- ((A ~<_ B /\ B ~< C) -> A ~< C)

Syntax hints:  -. wn 2   -> wi 3   \/ wo 239   /\ wa 240   e. wcel 1300  _Vcvv 2292   class class class wbr 3338   ~~ cen 5423   ~<_ cdom 5424   ~< csdm 5425

This theorem is referenced by:  pwuninel 5550  2pwuninel 5551  ondomon 6008  ondomcard 6009  cardmin 6012  alephsucdom 6028  infdif 8837  unpam2 14424  tarunpa 15235  intrtael 15256  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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