Theorem opth1 4221

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 3747	. . 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 4217	. . . . . . . 8 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩
6		id 19	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
7	5, 6	eleqtrid 2259	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
8		oprcl 3789	. . . . . . 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 111	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
11		prid1g 3687	. . . . 5 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷})
12	10, 11	syl 14	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ {𝐶, 𝐷})
13		eleq2 2234	. . . 4 ⊢ ({𝐴} = {𝐶, 𝐷} → (𝐶 ∈ {𝐴} ↔ 𝐶 ∈ {𝐶, 𝐷}))
14	12, 13	syl5ibrcom 156	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐶 ∈ {𝐴}))
15		elsni 3601	. . . 4 ⊢ (𝐶 ∈ {𝐴} → 𝐶 = 𝐴)
16	15	eqcomd 2176	. . 3 ⊢ (𝐶 ∈ {𝐴} → 𝐴 = 𝐶)
17	14, 16	syl6 33	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐴 = 𝐶))
18		dfopg 3763	. . . . 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 2249	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}})
21		elpri 3606	. . 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 713	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 703   = wceq 1348   ∈ wcel 2141  Vcvv 2730  {csn 3583  {cpr 3584  ⟨cop 3586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592

This theorem is referenced by:  opth  4222  dmsnopg  5082  funcnvsn  5243  oprabid  5885  pwle2  14031