Theorem opkth1g 4131

Description: Equality of the first member of a Kuratowski ordered pair, which holds regardless of the sethood of the second members. (Contributed by SF, 12-Jan-2015.)

Assertion

Ref	Expression
opkth1g	|- ((A e. V /\ <<A, B>> = <<C, D>>) -> A = C)

Proof of Theorem opkth1g

Step	Hyp	Ref	Expression
1		eqid 2353	. . . . 5 |- {C} = {C}
2	1	orci 379	. . . 4 |- ({C} = {C} \/ {C} = {C, D})
3		elopk 4130	. . . 4 |- ({C} e. <<C, D>> <-> ({C} = {C} \/ {C} = {C, D}))
4	2, 3	mpbir 200	. . 3 |- {C} e. <<C, D>>
5		eleq2 2414	. . . . 5 |- (<<A, B>> = <<C, D>> -> ({C} e. <<A, B>> <-> {C} e. <<C, D>>))
6	5	biimprd 214	. . . 4 |- (<<A, B>> = <<C, D>> -> ({C} e. <<C, D>> -> {C} e. <<A, B>>))
7		elopk 4130	. . . . 5 |- ({C} e. <<A, B>> <-> ({C} = {A} \/ {C} = {A, B}))
8		snidg 3759	. . . . . . 7 |- (A e. V -> A e. {A})
9		eleq2 2414	. . . . . . 7 |- ({C} = {A} -> (A e. {C} <-> A e. {A}))
10	8, 9	syl5ibrcom 213	. . . . . 6 |- (A e. V -> ({C} = {A} -> A e. {C}))
11		prid1g 3826	. . . . . . 7 |- (A e. V -> A e. {A, B})
12		eleq2 2414	. . . . . . 7 |- ({C} = {A, B} -> (A e. {C} <-> A e. {A, B}))
13	11, 12	syl5ibrcom 213	. . . . . 6 |- (A e. V -> ({C} = {A, B} -> A e. {C}))
14	10, 13	jaod 369	. . . . 5 |- (A e. V -> (({C} = {A} \/ {C} = {A, B}) -> A e. {C}))
15	7, 14	syl5bi 208	. . . 4 |- (A e. V -> ({C} e. <<A, B>> -> A e. {C}))
16	6, 15	sylan9r 639	. . 3 |- ((A e. V /\ <<A, B>> = <<C, D>>) -> ({C} e. <<C, D>> -> A e. {C}))
17	4, 16	mpi 16	. 2 |- ((A e. V /\ <<A, B>> = <<C, D>>) -> A e. {C})
18		elsncg 3756	. . 3 |- (A e. V -> (A e. {C} <-> A = C))
19	18	adantr 451	. 2 |- ((A e. V /\ <<A, B>> = <<C, D>>) -> (A e. {C} <-> A = C))
20	17, 19	mpbid 201	1 |- ((A e. V /\ <<A, B>> = <<C, D>>) -> A = C)

Syntax hints:   -> wi 4   <-> wb 176   \/ wo 357   /\ wa 358   = wceq 1642   e. wcel 1710  {csn 3738  {cpr 3739  <<copk 4058

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-sn 3742  df-pr 3743  df-opk 4059

This theorem is referenced by:  opkthg  4132