Theorem opth1 5384

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 5377	. . 3 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩
4		id 22	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
5	3, 4	eleqtrid 2845	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
6	1	sneqr 4768	. . . 4 ⊢ ({𝐴} = {𝐶} → 𝐴 = 𝐶)
7	6	a1i 11	. . 3 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶} → 𝐴 = 𝐶))
8		oprcl 4827	. . . . . . 7 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
9	8	simpld 494	. . . . . 6 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
10		prid1g 4693	. . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷})
11	9, 10	syl 17	. . . . 5 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ {𝐶, 𝐷})
12		eleq2 2827	. . . . 5 ⊢ ({𝐴} = {𝐶, 𝐷} → (𝐶 ∈ {𝐴} ↔ 𝐶 ∈ {𝐶, 𝐷}))
13	11, 12	syl5ibrcom 246	. . . 4 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐶 ∈ {𝐴}))
14		elsni 4575	. . . . 5 ⊢ (𝐶 ∈ {𝐴} → 𝐶 = 𝐴)
15	14	eqcomd 2744	. . . 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 4799	. . . . . 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 2841	. . . 4 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}})
21		elpri 4580	. . . 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 856	. 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 843   = wceq 1539   ∈ wcel 2108  Vcvv 3422  {csn 4558  {cpr 4560  ⟨cop 4564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-un 3888  df-nul 4254  df-if 4457  df-sn 4559  df-pr 4561  df-op 4565

This theorem is referenced by:  opth  5385  dmsnopg  6105  funcnvsn  6468  oprabidw  7286  oprabid  7287  seqomlem2  8252  unxpdomlem3  8958  dfac5lem4  9813  dcomex  10134  canthwelem  10337  uzrdgfni  13606  fnpr2ob  17186  gsum2d2  19490  poimirlem9  35713  ichnreuop  44812  ichreuopeq  44813