Theorem ensdomtr 4405

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 4355	. . . . . . . 8 |- ((A ~~ B /\ B ~<_ C) -> A ~<_ C)
2	1	ex 373	. . . . . . 7 |- (A ~~ B -> (B ~<_ C -> A ~<_ C))
3	2	adantl 388	. . . . . 6 |- ((B e. V /\ A ~~ B) -> (B ~<_ C -> A ~<_ C))
4		ensymg 4346	. . . . . . . . 9 |- (B e. V -> (A ~~ B -> B ~~ A))
5	4	imp 350	. . . . . . . 8 |- ((B e. V /\ A ~~ B) -> B ~~ A)
6		entrt 4349	. . . . . . . . 9 |- ((B ~~ A /\ A ~~ C) -> B ~~ C)
7	6	ex 373	. . . . . . . 8 |- (B ~~ A -> (A ~~ C -> B ~~ C))
8	5, 7	syl 10	. . . . . . 7 |- ((B e. V /\ A ~~ B) -> (A ~~ C -> B ~~ C))
9	8	con3d 95	. . . . . 6 |- ((B e. V /\ A ~~ B) -> (-. B ~~ C -> -. A ~~ C))
10	3, 9	anim12d 556	. . . . 5 |- ((B e. V /\ A ~~ B) -> ((B ~<_ C /\ -. B ~~ C) -> (A ~<_ C /\ -. A ~~ C)))
11		brsdom 4317	. . . . 5 |- (B ~< C <-> (B ~<_ C /\ -. B ~~ C))
12		brsdom 4317	. . . . 5 |- (A ~< C <-> (A ~<_ C /\ -. A ~~ C))
13	10, 11, 12	3imtr4g 551	. . . 4 |- ((B e. V /\ A ~~ B) -> (B ~< C -> A ~< C))
14	13	ex 373	. . 3 |- (B e. V -> (A ~~ B -> (B ~< C -> A ~< C)))
15	14	imp3a 361	. 2 |- (B e. V -> ((A ~~ B /\ B ~< C) -> A ~< C))
16		relsdom 4310	. . . . . 6 |- Rel ~<
17	16	brrelexi 3170	. . . . 5 |- (B ~< C -> B e. V)
18	17	con3i 98	. . . 4 |- (-. B e. V -> -. B ~< C)
19	18	pm2.21d 78	. . 3 |- (-. B e. V -> (B ~< C -> A ~< C))
20	19	adantld 390	. 2 |- (-. B e. V -> ((A ~~ B /\ B ~< C) -> A ~< C))
21	15, 20	pm2.61i 126	1 |- ((A ~~ B /\ B ~< C) -> A ~< C)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 1105  Vcvv 1786   class class class wbr 2587   ~~ cen 4302   ~<_ cdom 4303   ~< csdm 4304

This theorem is referenced by:  sdomen1 4415  isfinite2 4475  pm54.43 4498  alephordi 4797  resdomq 7444  aleph1re 7445  infdif 7462  aleph1irr 7471

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-4 951  ax-5 952  ax-6 953  ax-7 954  ax-gen 955  ax-8 1101  ax-9 1102  ax-10 1103  ax-12 1104  ax-13 1107  ax-14 1108  ax-11 1180  ax-17 1190  ax-16 1194  ax-11o 1202  ax-ext 1436  ax-rep 2661  ax-sep 2671  ax-nul 2678  ax-pow 2710  ax-pr 2747  ax-un 2830

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 774  df-ex 957  df-sb 1155  df-eu 1359  df-mo 1360  df-clab 1441  df-cleq 1446  df-clel 1449  df-ne 1563  df-rex 1626  df-v 1787  df-dif 2020  df-un 2021  df-in 2022  df-ss 2024  df-nul 2252  df-pw 2373  df-sn 2383  df-pr 2384  df-op 2387  df-uni 2472  df-br 2588  df-opab 2635  df-id 2797  df-xp 3147  df-rel 3148  df-cnv 3149  df-co 3150  df-dm 3151  df-rn 3152  df-res 3153  df-ima 3154  df-fun 3155  df-fn 3156  df-f 3157  df-f1 3158  df-fo 3159  df-f1o 3160  df-er 4199  df-en 4305  df-dom 4306  df-sdom 4307

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