Theorem opth1 5450

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		opth1.2	. . . 4 ⊢ 𝐵 ∈ V
3	1, 2	opi1 5443	. . 3 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩
4		id 22	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
5	3, 4	eleqtrid 2840	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
6	1	sneqr 4816	. . . 4 ⊢ ({𝐴} = {𝐶} → 𝐴 = 𝐶)
7	6	a1i 11	. . 3 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶} → 𝐴 = 𝐶))
8		oprcl 4875	. . . . . . 7 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
9	8	simpld 494	. . . . . 6 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
10		prid1g 4736	. . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷})
11	9, 10	syl 17	. . . . 5 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ {𝐶, 𝐷})
12		eleq2 2823	. . . . 5 ⊢ ({𝐴} = {𝐶, 𝐷} → (𝐶 ∈ {𝐴} ↔ 𝐶 ∈ {𝐶, 𝐷}))
13	11, 12	syl5ibrcom 247	. . . 4 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐶 ∈ {𝐴}))
14		elsni 4618	. . . . 5 ⊢ (𝐶 ∈ {𝐴} → 𝐶 = 𝐴)
15	14	eqcomd 2741	. . . 4 ⊢ (𝐶 ∈ {𝐴} → 𝐴 = 𝐶)
16	13, 15	syl6 35	. . 3 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐴 = 𝐶))
17		id 22	. . . . 5 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
18		dfopg 4847	. . . . . 6 ⊢ ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
19	8, 18	syl 17	. . . . 5 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
20	17, 19	eleqtrd 2836	. . . 4 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}})
21		elpri 4625	. . . 4 ⊢ ({𝐴} ∈ {{𝐶}, {𝐶, 𝐷}} → ({𝐴} = {𝐶} ∨ {𝐴} = {𝐶, 𝐷}))
22	20, 21	syl 17	. . 3 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶} ∨ {𝐴} = {𝐶, 𝐷}))
23	7, 16, 22	mpjaod 860	. 2 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
24	5, 23	syl 17	1 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2108  Vcvv 3459  {csn 4601  {cpr 4603  ⟨cop 4607

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-v 3461  df-dif 3929  df-un 3931  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608

This theorem is referenced by:  opth  5451  dmsnopg  6202  funcnvsn  6585  oprabidw  7434  oprabid  7435  seqomlem2  8463  unxpdomlem3  9258  dfac5lem4  10138  dfac5lem4OLD  10140  dcomex  10459  canthwelem  10662  uzrdgfni  13974  fnpr2ob  17570  gsum2d2  19953  noseqrdgfn  28229  poimirlem9  37599  ichnreuop  47434  ichreuopeq  47435  diag1f1lem  49134  idfudiag1bas  49324