Theorem opth1 4230

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 3756	. . 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 4226	. . . . . . . 8 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩
6		id 19	. . . . . . . 8 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
7	5, 6	eleqtrid 2264	. . . . . . 7 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
8		oprcl 3798	. . . . . . 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 112	. . . . 5 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
11		prid1g 3693	. . . . 5 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷})
12	10, 11	syl 14	. . . 4 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 ∈ {𝐶, 𝐷})
13		eleq2 2239	. . . 4 ⊢ ({𝐴} = {𝐶, 𝐷} → (𝐶 ∈ {𝐴} ↔ 𝐶 ∈ {𝐶, 𝐷}))
14	12, 13	syl5ibrcom 157	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐶 ∈ {𝐴}))
15		elsni 3607	. . . 4 ⊢ (𝐶 ∈ {𝐴} → 𝐶 = 𝐴)
16	15	eqcomd 2181	. . 3 ⊢ (𝐶 ∈ {𝐴} → 𝐴 = 𝐶)
17	14, 16	syl6 33	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐴 = 𝐶))
18		dfopg 3772	. . . . 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 2254	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}})
21		elpri 3612	. . 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 718	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 708   = wceq 1353   ∈ wcel 2146  Vcvv 2735  {csn 3589  {cpr 3590  ⟨cop 3592

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-v 2737  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598

This theorem is referenced by:  opth  4231  dmsnopg  5092  funcnvsn  5253  oprabid  5897  pwle2  14288