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

Step	Hyp	Ref	Expression
1		eqid 2353	. . . . 5 ⊢ {C} = {C}
2	1	orci 379	. . . 4 ⊢ ({C} = {C} ∨ {C} = {C, D})
3		elopk 4129	. . . 4 ⊢ ({C} ∈ ⟪C, D⟫ ↔ ({C} = {C} ∨ {C} = {C, D}))
4	2, 3	mpbir 200	. . 3 ⊢ {C} ∈ ⟪C, D⟫
5		eleq2 2414	. . . . 5 ⊢ (⟪A, B⟫ = ⟪C, D⟫ → ({C} ∈ ⟪A, B⟫ ↔ {C} ∈ ⟪C, D⟫))
6	5	biimprd 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 ∈ V → A ∈ {A})
9		eleq2 2414	. . . . . . 7 ⊢ ({C} = {A} → (A ∈ {C} ↔ A ∈ {A}))
10	8, 9	syl5ibrcom 213	. . . . . 6 ⊢ (A ∈ V → ({C} = {A} → A ∈ {C}))
11		prid1g 3825	. . . . . . 7 ⊢ (A ∈ V → A ∈ {A, B})
12		eleq2 2414	. . . . . . 7 ⊢ ({C} = {A, B} → (A ∈ {C} ↔ A ∈ {A, B}))
13	11, 12	syl5ibrcom 213	. . . . . 6 ⊢ (A ∈ V → ({C} = {A, B} → A ∈ {C}))
14	10, 13	jaod 369	. . . . 5 ⊢ (A ∈ V → (({C} = {A} ∨ {C} = {A, B}) → A ∈ {C}))
15	7, 14	syl5bi 208	. . . 4 ⊢ (A ∈ V → ({C} ∈ ⟪A, B⟫ → A ∈ {C}))
16	6, 15	sylan9r 639	. . 3 ⊢ ((A ∈ V ∧ ⟪A, B⟫ = ⟪C, D⟫) → ({C} ∈ ⟪C, D⟫ → A ∈ {C}))
17	4, 16	mpi 16	. 2 ⊢ ((A ∈ V ∧ ⟪A, B⟫ = ⟪C, D⟫) → A ∈ {C})
18		elsncg 3755	. . 3 ⊢ (A ∈ V → (A ∈ {C} ↔ A = C))
19	18	adantr 451	. 2 ⊢ ((A ∈ V ∧ ⟪A, B⟫ = ⟪C, D⟫) → (A ∈ {C} ↔ A = C))
20	17, 19	mpbid 201	1 ⊢ ((A ∈ V ∧ ⟪A, B⟫ = ⟪C, D⟫) → A = C)

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-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 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