Theorem opelxp1 3211

Description: The first member of an ordered pair of classes in a cross product belongs to first cross product argument.

Assertion

Ref	Expression
opelxp1	|- (<.A, B>. e. (C X. D) -> A e. C)

Proof of Theorem opelxp1

Step	Hyp	Ref	Expression
1		elxp 3208	. 2 |- (<.A, B>. e. (C X. D) <-> E.xE.y(<.A, B>. = <.x, y>. /\ (x e. C /\ y e. D)))
2		visset 1816	. . . . . . . 8 |- x e. V
3	2	opth1 2792	. . . . . . 7 |- (<.x, y>. = <.A, B>. -> x = A)
4	3	eqcoms 1481	. . . . . 6 |- (<.A, B>. = <.x, y>. -> x = A)
5	4	eleq1d 1543	. . . . 5 |- (<.A, B>. = <.x, y>. -> (x e. C <-> A e. C))
6	5	biimpa 418	. . . 4 |- ((<.A, B>. = <.x, y>. /\ x e. C) -> A e. C)
7	6	adantrr 397	. . 3 |- ((<.A, B>. = <.x, y>. /\ (x e. C /\ y e. D)) -> A e. C)
8	7	19.23aivv 1298	. 2 |- (E.xE.y(<.A, B>. = <.x, y>. /\ (x e. C /\ y e. D)) -> A e. C)
9	1, 8	sylbi 199	1 |- (<.A, B>. e. (C X. D) -> A e. C)

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  E.wex 982  <.cop 2415   X. cxp 3174

This theorem is referenced by:  otelxp1 3212  brrelex 3213  opelxp 3220  opelxpex2 3285  eqop 4110  vcoprnelem 8193  vcex 8195

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-opab 2672  df-xp 3190

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