Theorem opelxpex2 4119

Description: The second member of an ordered pair of classes in a cross product exists when the order pair doesn't belong to _I. (The proof was shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
opelxpex2	|- (<.A, B>. e. ((C X. D) \ _I ) -> B e. _V)

Proof of Theorem opelxpex2

Step	Hyp	Ref	Expression
1		eldif 2609	. 2 |- (<.A, B>. e. ((C X. D) \ _I ) <-> (<.A, B>. e. (C X. D) /\ -. <.A, B>. e. _I ))
2		opelxp1 4026	. . . 4 |- (<.A, B>. e. (C X. D) -> A e. C)
3		brprc 3386	. . . . . . 7 |- (-. B e. _V -> (A _I B <-> A _I A))
4		df-br 3339	. . . . . . 7 |- (A _I B <-> <.A, B>. e. _I )
5	3, 4	syl5bbr 593	. . . . . 6 |- (-. B e. _V -> (<.A, B>. e. _I <-> A _I A))
6		ididg 4117	. . . . . 6 |- (A e. C -> A _I A)
7	5, 6	syl5cbir 228	. . . . 5 |- (A e. C -> (-. B e. _V -> <.A, B>. e. _I ))
8	7	con1d 109	. . . 4 |- (A e. C -> (-. <.A, B>. e. _I -> B e. _V))
9	2, 8	syl 12	. . 3 |- (<.A, B>. e. (C X. D) -> (-. <.A, B>. e. _I -> B e. _V))
10	9	imp 377	. 2 |- ((<.A, B>. e. (C X. D) /\ -. <.A, B>. e. _I ) -> B e. _V)
11	1, 10	sylbi 216	1 |- (<.A, B>. e. ((C X. D) \ _I ) -> B e. _V)

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   e. wcel 1300  _Vcvv 2292   \ cdif 2590  <.cop 3046   class class class wbr 3338   _I cid 3582   X. cxp 3984

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-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

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  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-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-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001

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