Theorem sndw 14428

Description: If A is a part of B and B a part of C and A is equipotent to C then A is equipotent to B. The art of sandwich applied to set theory.

Assertion

Ref	Expression
sndw	|- ((A C_ B /\ B C_ C /\ C e. _V) -> (A ~~ C -> A ~~ B))

Proof of Theorem sndw

Step	Hyp	Ref	Expression
1		relen 5431	. . . . . . . . . 10 |- Rel ~~
2	1	brrelexi 4029	. . . . . . . . 9 |- (A ~~ C -> A e. _V)
3	2	3ad2ant3 899	. . . . . . . 8 |- ((B C_ C /\ C e. _V /\ A ~~ C) -> A e. _V)
4		ssdomg 5467	. . . . . . . 8 |- (A e. _V -> (A C_ B -> A ~<_ B))
5	3, 4	syl 12	. . . . . . 7 |- ((B C_ C /\ C e. _V /\ A ~~ C) -> (A C_ B -> A ~<_ B))
6	5	3exp 1066	. . . . . 6 |- (B C_ C -> (C e. _V -> (A ~~ C -> (A C_ B -> A ~<_ B))))
7	6	com4r 45	. . . . 5 |- (A C_ B -> (B C_ C -> (C e. _V -> (A ~~ C -> A ~<_ B))))
8	7	3imp1 1081	. . . 4 |- (((A C_ B /\ B C_ C /\ C e. _V) /\ A ~~ C) -> A ~<_ B)
9		ssdom2g 5468	. . . . . . . 8 |- (C e. _V -> (B C_ C -> B ~<_ C))
10	9	impcom 378	. . . . . . 7 |- ((B C_ C /\ C e. _V) -> B ~<_ C)
11	10	3adant1 894	. . . . . 6 |- ((A C_ B /\ B C_ C /\ C e. _V) -> B ~<_ C)
12	11	adantr 425	. . . . 5 |- (((A C_ B /\ B C_ C /\ C e. _V) /\ A ~~ C) -> B ~<_ C)
13		ensymg 5470	. . . . . . . 8 |- (C e. _V -> (A ~~ C -> C ~~ A))
14	13	3ad2ant3 899	. . . . . . 7 |- ((A C_ B /\ B C_ C /\ C e. _V) -> (A ~~ C -> C ~~ A))
15		endom 5444	. . . . . . 7 |- (C ~~ A -> C ~<_ A)
16	14, 15	syl6 25	. . . . . 6 |- ((A C_ B /\ B C_ C /\ C e. _V) -> (A ~~ C -> C ~<_ A))
17	16	imp 377	. . . . 5 |- (((A C_ B /\ B C_ C /\ C e. _V) /\ A ~~ C) -> C ~<_ A)
18		domtr 5474	. . . . 5 |- ((B ~<_ C /\ C ~<_ A) -> B ~<_ A)
19	12, 17, 18	syl11anc 524	. . . 4 |- (((A C_ B /\ B C_ C /\ C e. _V) /\ A ~~ C) -> B ~<_ A)
20	8, 19	jca 310	. . 3 |- (((A C_ B /\ B C_ C /\ C e. _V) /\ A ~~ C) -> (A ~<_ B /\ B ~<_ A))
21	20	ex 402	. 2 |- ((A C_ B /\ B C_ C /\ C e. _V) -> (A ~~ C -> (A ~<_ B /\ B ~<_ A)))
22		sbth 5520	. 2 |- ((A ~<_ B /\ B ~<_ A) -> A ~~ B)
23	21, 22	syl6 25	1 |- ((A C_ B /\ B C_ C /\ C e. _V) -> (A ~~ C -> A ~~ B))

Syntax hints:   -> wi 3   /\ wa 240   /\ w3a 858   e. wcel 1300  _Vcvv 2292   C_ wss 2593   class class class wbr 3338   ~~ cen 5423   ~<_ cdom 5424

This theorem is referenced by:  sndw2 14429  intartar 15255

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

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