Theorem opth1 2786

Description: Equality of the first members of equal ordered pairs, which holds whether or not the second members are sets.

Hypothesis

Ref	Expression
opth1.1	|- A e. V

Assertion

Ref	Expression
opth1	|- (<.A, B>. = <.C, D>. -> A = C)

Proof of Theorem opth1

Step	Hyp	Ref	Expression
1		opi1 2784	. . . . 5 |- {C} e. <.C, D>.
2		eleq2 1535	. . . . 5 |- (<.A, B>. = <.C, D>. -> ({C} e. <.A, B>. <-> {C} e. <.C, D>.))
3	1, 2	mpbiri 194	. . . 4 |- (<.A, B>. = <.C, D>. -> {C} e. <.A, B>.)
4		snex 2750	. . . . 5 |- {C} e. V
5	4	elop 2783	. . . 4 |- ({C} e. <.A, B>. <-> ({C} = {A} \/ {C} = {A, B}))
6	3, 5	sylib 198	. . 3 |- (<.A, B>. = <.C, D>. -> ({C} = {A} \/ {C} = {A, B}))
7		opth1.1	. . . . . 6 |- A e. V
8	7	snid 2435	. . . . 5 |- A e. {A}
9		eleq2 1535	. . . . 5 |- ({C} = {A} -> (A e. {C} <-> A e. {A}))
10	8, 9	mpbiri 194	. . . 4 |- ({C} = {A} -> A e. {C})
11	7	pri1 2450	. . . . 5 |- A e. {A, B}
12		eleq2 1535	. . . . 5 |- ({C} = {A, B} -> (A e. {C} <-> A e. {A, B}))
13	11, 12	mpbiri 194	. . . 4 |- ({C} = {A, B} -> A e. {C})
14	10, 13	jaoi 341	. . 3 |- (({C} = {A} \/ {C} = {A, B}) -> A e. {C})
15	6, 14	syl 10	. 2 |- (<.A, B>. = <.C, D>. -> A e. {C})
16	7	elsnc 2431	. 2 |- (A e. {C} <-> A = C)
17	15, 16	sylib 198	1 |- (<.A, B>. = <.C, D>. -> A = C)

Syntax hints:   -> wi 3   \/ wo 222   = wceq 956   e. wcel 958  Vcvv 1811  {csn 2409  {cpr 2410  <.cop 2411

This theorem is referenced by:  opth 2787  oteqex 2799  opelxp1 3205

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416

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