Theorem ssref 15492

Description: A subcover is a refinement of the original cover.

Hypotheses

Ref	Expression
ssref.1	|- X = U.A
ssref.2	|- Y = U.B

Assertion

Ref	Expression
ssref	|- ((A e. C /\ A C_ B /\ X = Y) -> BRefA)

Proof of Theorem ssref

Step	Hyp	Ref	Expression
1		ssref.2	. . . 4 |- Y = U.B
2		ssref.1	. . . 4 |- X = U.A
3	1, 2	isref 15488	. . 3 |- (A e. C -> (BRefA <-> (Y = X /\ A.x e. A E.y e. B x C_ y)))
4	3	3ad2ant1 897	. 2 |- ((A e. C /\ A C_ B /\ X = Y) -> (BRefA <-> (Y = X /\ A.x e. A E.y e. B x C_ y)))
5		eqcom 1886	. . . 4 |- (X = Y <-> Y = X)
6	5	biimpi 168	. . 3 |- (X = Y -> Y = X)
7	6	3ad2ant3 899	. 2 |- ((A e. C /\ A C_ B /\ X = Y) -> Y = X)
8		ssel2 2616	. . . . 5 |- ((A C_ B /\ x e. A) -> x e. B)
9	8	3ad2antl2 1039	. . . 4 |- (((A e. C /\ A C_ B /\ X = Y) /\ x e. A) -> x e. B)
10		ssid 2634	. . . . 5 |- x C_ x
11	10	a1i 8	. . . 4 |- (((A e. C /\ A C_ B /\ X = Y) /\ x e. A) -> x C_ x)
12		sseq2 2639	. . . . 5 |- (y = x -> (x C_ y <-> x C_ x))
13	12	rcla4ev 2381	. . . 4 |- ((x e. B /\ x C_ x) -> E.y e. B x C_ y)
14	9, 11, 13	syl11anc 524	. . 3 |- (((A e. C /\ A C_ B /\ X = Y) /\ x e. A) -> E.y e. B x C_ y)
15	14	r19.21aiva 2176	. 2 |- ((A e. C /\ A C_ B /\ X = Y) -> A.x e. A E.y e. B x C_ y)
16	4, 7, 15	mpbir2and 802	1 |- ((A e. C /\ A C_ B /\ X = Y) -> BRefA)

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   /\ w3a 858   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   C_ wss 2593  U.cuni 3177   class class class wbr 3338  Refcref 15458

This theorem is referenced by:  refssfne 15504

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-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-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-xp 4000  df-rel 4001  df-ref 15464

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