Theorem opth1 5418

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 5411	. . 3 ⊢ {𝐴} ∈ ⟨𝐴, 𝐵⟩
4		id 22	. . 3 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
5	3, 4	eleqtrid 2834	. 2 ⊢ (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
6	1	sneqr 4791	. . . 4 ⊢ ({𝐴} = {𝐶} → 𝐴 = 𝐶)
7	6	a1i 11	. . 3 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶} → 𝐴 = 𝐶))
8		oprcl 4850	. . . . . . 7 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
9	8	simpld 494	. . . . . 6 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
10		prid1g 4712	. . . . . 6 ⊢ (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷})
11	9, 10	syl 17	. . . . 5 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ {𝐶, 𝐷})
12		eleq2 2817	. . . . 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 4594	. . . . 5 ⊢ (𝐶 ∈ {𝐴} → 𝐶 = 𝐴)
15	14	eqcomd 2735	. . . 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 4822	. . . . . 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 2830	. . . 4 ⊢ ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}})
21		elpri 4601	. . . 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 2109  Vcvv 3436  {csn 4577  {cpr 4579  ⟨cop 4583

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 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

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 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3438  df-dif 3906  df-un 3908  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584

This theorem is referenced by:  opth  5419  dmsnopg  6162  funcnvsn  6532  oprabidw  7380  oprabid  7381  seqomlem2  8373  unxpdomlem3  9147  dfac5lem4  10020  dfac5lem4OLD  10022  dcomex  10341  canthwelem  10544  uzrdgfni  13865  fnpr2ob  17462  gsum2d2  19853  noseqrdgfn  28205  poimirlem9  37613  ichnreuop  47460  ichreuopeq  47461  diag1f1lem  49295  idfudiag1bas  49513