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Theorem opkth1g 4130
 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 V A, B⟫ = ⟪C, D⟫) → A = C)

Proof of Theorem opkth1g
StepHypRef Expression
1 eqid 2353 . . . . 5 {C} = {C}
21orci 379 . . . 4 ({C} = {C} {C} = {C, D})
3 elopk 4129 . . . 4 ({C} C, D⟫ ↔ ({C} = {C} {C} = {C, D}))
42, 3mpbir 200 . . 3 {C} C, D
5 eleq2 2414 . . . . 5 (⟪A, B⟫ = ⟪C, D⟫ → ({C} A, B⟫ ↔ {C} C, D⟫))
65biimprd 214 . . . 4 (⟪A, B⟫ = ⟪C, D⟫ → ({C} C, D⟫ → {C} A, B⟫))
7 elopk 4129 . . . . 5 ({C} A, B⟫ ↔ ({C} = {A} {C} = {A, B}))
8 snidg 3758 . . . . . . 7 (A VA {A})
9 eleq2 2414 . . . . . . 7 ({C} = {A} → (A {C} ↔ A {A}))
108, 9syl5ibrcom 213 . . . . . 6 (A V → ({C} = {A} → A {C}))
11 prid1g 3825 . . . . . . 7 (A VA {A, B})
12 eleq2 2414 . . . . . . 7 ({C} = {A, B} → (A {C} ↔ A {A, B}))
1311, 12syl5ibrcom 213 . . . . . 6 (A V → ({C} = {A, B} → A {C}))
1410, 13jaod 369 . . . . 5 (A V → (({C} = {A} {C} = {A, B}) → A {C}))
157, 14syl5bi 208 . . . 4 (A V → ({C} A, B⟫ → A {C}))
166, 15sylan9r 639 . . 3 ((A V A, B⟫ = ⟪C, D⟫) → ({C} C, D⟫ → A {C}))
174, 16mpi 16 . 2 ((A V A, B⟫ = ⟪C, D⟫) → A {C})
18 elsncg 3755 . . 3 (A V → (A {C} ↔ A = C))
1918adantr 451 . 2 ((A V A, B⟫ = ⟪C, D⟫) → (A {C} ↔ A = C))
2017, 19mpbid 201 1 ((A V A, B⟫ = ⟪C, D⟫) → A = C)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∨ wo 357   ∧ wa 358   = wceq 1642   ∈ wcel 1710  {csn 3737  {cpr 3738  ⟪copk 4057 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 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 4078  ax-sn 4087 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 2478  df-ne 2518  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-ss 3259  df-nul 3551  df-sn 3741  df-pr 3742  df-opk 4058 This theorem is referenced by:  opkthg  4131
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