Theorem opth1 3973

Description: Equality of the first members of equal ordered pairs. (Contributed by NM, 28-May-2008.) (Revised by Mario Carneiro, 26-Apr-2015.)

Hypotheses

Ref	Expression
opth1.1	⊢ 𝐴 ∈ V
opth1.2	⊢ 𝐵 ∈ V

Assertion

Ref	Expression
opth1	⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)

Proof of Theorem opth1

Step	Hyp	Ref	Expression
1		opth1.1	. . . 4 ⊢ 𝐴 ∈ V
2	1	sneqr 3531	. . 3 ⊢ ({𝐴} = {𝐶} → 𝐴 = 𝐶)
3	2	a1i 9	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶} → 𝐴 = 𝐶))
4		opth1.2	. . . . . . . . 9 ⊢ 𝐵 ∈ V
5	1, 4	opi1 3969	. . . . . . . 8 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩
6		id 19	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
7	5, 6	syl5eleq 2126	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
8		oprcl 3573	. . . . . . 7 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
9	7, 8	syl 14	. . . . . 6 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
10	9	simpld 105	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
11		prid1g 3474	. . . . 5 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷})
12	10, 11	syl 14	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ {𝐶, 𝐷})
13		eleq2 2101	. . . 4 ⊢ ({𝐴} = {𝐶, 𝐷} → (𝐶 ∈ {𝐴} ↔ 𝐶 ∈ {𝐶, 𝐷}))
14	12, 13	syl5ibrcom 146	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐶 ∈ {𝐴}))
15		elsni 3393	. . . 4 ⊢ (𝐶 ∈ {𝐴} → 𝐶 = 𝐴)
16	15	eqcomd 2045	. . 3 ⊢ (𝐶 ∈ {𝐴} → 𝐴 = 𝐶)
17	14, 16	syl6 29	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐴 = 𝐶))
18		dfopg 3547	. . . . 5 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
19	7, 8, 18	3syl 17	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
20	7, 19	eleqtrd 2116	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}})
21		elpri 3398	. . 3 ⊢ ({𝐴} ∈ {{𝐶}, {𝐶, 𝐷}} → ({𝐴} = {𝐶} ∨ {𝐴} = {𝐶, 𝐷}))
22	20, 21	syl 14	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶} ∨ {𝐴} = {𝐶, 𝐷}))
23	3, 17, 22	mpjaod 638	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 97   ∨ wo 629   = wceq 1243   ∈ wcel 1393  Vcvv 2557  {csn 3375  {cpr 3376  ⟨cop 3378

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384

This theorem is referenced by:  opth  3974  dmsnopg  4792  funcnvsn  4945  oprabid  5537