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Theorem opth1 5071
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
StepHypRef Expression
1 opth1.1 . . . 4 𝐴 ∈ V
2 opth1.2 . . . 4 𝐵 ∈ V
31, 2opi1 5064 . . 3 {𝐴} ∈ ⟨𝐴, 𝐵
4 id 22 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
53, 4syl5eleq 2855 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
61sneqr 4502 . . . 4 ({𝐴} = {𝐶} → 𝐴 = 𝐶)
76a1i 11 . . 3 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶} → 𝐴 = 𝐶))
8 oprcl 4563 . . . . . . 7 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
98simpld 476 . . . . . 6 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
10 prid1g 4429 . . . . . 6 (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷})
119, 10syl 17 . . . . 5 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ {𝐶, 𝐷})
12 eleq2 2838 . . . . 5 ({𝐴} = {𝐶, 𝐷} → (𝐶 ∈ {𝐴} ↔ 𝐶 ∈ {𝐶, 𝐷}))
1311, 12syl5ibrcom 237 . . . 4 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐶 ∈ {𝐴}))
14 elsni 4331 . . . . 5 (𝐶 ∈ {𝐴} → 𝐶 = 𝐴)
1514eqcomd 2776 . . . 4 (𝐶 ∈ {𝐴} → 𝐴 = 𝐶)
1613, 15syl6 35 . . 3 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐴 = 𝐶))
17 id 22 . . . . 5 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
18 dfopg 4535 . . . . . 6 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
198, 18syl 17 . . . . 5 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
2017, 19eleqtrd 2851 . . . 4 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}})
21 elpri 4335 . . . 4 ({𝐴} ∈ {{𝐶}, {𝐶, 𝐷}} → ({𝐴} = {𝐶} ∨ {𝐴} = {𝐶, 𝐷}))
2220, 21syl 17 . . 3 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶} ∨ {𝐴} = {𝐶, 𝐷}))
237, 16, 22mpjaod 840 . 2 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
245, 23syl 17 1 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → 𝐴 = 𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wo 826   = wceq 1630  wcel 2144  Vcvv 3349  {csn 4314  {cpr 4316  cop 4320
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321
This theorem is referenced by:  opth  5072  dmsnopg  5748  funcnvsn  6079  oprabid  6821  seqomlem2  7698  unxpdomlem3  8321  dfac5lem4  9148  dcomex  9470  canthwelem  9673  uzrdgfni  12964  gsum2d2  18579  poimirlem9  33744
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