Theorem ensdomtr 4558

Description: Transitivity of equinumerosity and strict dominance.

Assertion

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

Proof of Theorem ensdomtr

Step	Hyp	Ref	Expression
1		endomtr 4507	. . . . . . 7 |- ((A ~~ B /\ B ~<_ C) -> A ~<_ C)
2	1	ex 371	. . . . . 6 |- (A ~~ B -> (B ~<_ C -> A ~<_ C))
3	2	adantl 388	. . . . 5 |- ((B e. V /\ A ~~ B) -> (B ~<_ C -> A ~<_ C))
4		ensymg 4498	. . . . . . . 8 |- (B e. V -> (A ~~ B -> B ~~ A))
5	4	imp 348	. . . . . . 7 |- ((B e. V /\ A ~~ B) -> B ~~ A)
6		entr 4501	. . . . . . . 8 |- ((B ~~ A /\ A ~~ C) -> B ~~ C)
7	6	ex 371	. . . . . . 7 |- (B ~~ A -> (A ~~ C -> B ~~ C))
8	5, 7	syl 10	. . . . . 6 |- ((B e. V /\ A ~~ B) -> (A ~~ C -> B ~~ C))
9	8	con3d 95	. . . . 5 |- ((B e. V /\ A ~~ B) -> (-. B ~~ C -> -. A ~~ C))
10	3, 9	anim12d 560	. . . 4 |- ((B e. V /\ A ~~ B) -> ((B ~<_ C /\ -. B ~~ C) -> (A ~<_ C /\ -. A ~~ C)))
11		brsdom 4468	. . . 4 |- (B ~< C <-> (B ~<_ C /\ -. B ~~ C))
12		brsdom 4468	. . . 4 |- (A ~< C <-> (A ~<_ C /\ -. A ~~ C))
13	10, 11, 12	3imtr4g 555	. . 3 |- ((B e. V /\ A ~~ B) -> (B ~< C -> A ~< C))
14	13	expimpd 373	. 2 |- (B e. V -> ((A ~~ B /\ B ~< C) -> A ~< C))
15		relsdom 4461	. . . . . 6 |- Rel ~<
16	15	brrelexi 3265	. . . . 5 |- (B ~< C -> B e. V)
17	16	con3i 98	. . . 4 |- (-. B e. V -> -. B ~< C)
18	17	pm2.21d 78	. . 3 |- (-. B e. V -> (B ~< C -> A ~< C))
19	18	adantld 390	. 2 |- (-. B e. V -> ((A ~~ B /\ B ~< C) -> A ~< C))
20	14, 19	pm2.61i 124	1 |- ((A ~~ B /\ B ~< C) -> A ~< C)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 221   e. wcel 990  Vcvv 1849   class class class wbr 2669   ~~ cen 4451   ~<_ cdom 4452   ~< csdm 4453

This theorem is referenced by:  sdomen1 4568  isfinite2 4633  pm54.43 4656  alephordi 4963  resdomq 7675  aleph1re 7676  infdif 7693  infpss 7699  aleph1irr 7703  top2ind 10684

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 994  ax-gen 995  ax-8 996  ax-9 997  ax-10 998  ax-11 999  ax-12 1000  ax-13 1001  ax-14 1002  ax-17 1003  ax-4 1005  ax-5o 1007  ax-6o 1010  ax-9o 1155  ax-10o 1173  ax-16 1243  ax-11o 1251  ax-ext 1494  ax-rep 2744  ax-sep 2754  ax-pow 2794  ax-pr 2832  ax-un 2920

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-3an 780  df-ex 1013  df-sb 1205  df-eu 1415  df-mo 1416  df-clab 1500  df-cleq 1505  df-clel 1508  df-ne 1624  df-rex 1688  df-v 1850  df-dif 2093  df-un 2094  df-in 2095  df-ss 2097  df-nul 2325  df-pw 2447  df-sn 2457  df-pr 2458  df-op 2461  df-uni 2552  df-br 2670  df-opab 2718  df-id 2889  df-xp 3239  df-rel 3240  df-cnv 3241  df-co 3242  df-dm 3243  df-rn 3244  df-res 3245  df-ima 3246  df-fun 3247  df-fn 3248  df-f 3249  df-f1 3250  df-fo 3251  df-f1o 3252  df-er 4345  df-en 4455  df-dom 4456  df-sdom 4457

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