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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 2856 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
61sneqr 4504 . . . 4 ({𝐴} = {𝐶} → 𝐴 = 𝐶)
76a1i 11 . . 3 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶} → 𝐴 = 𝐶))
8 oprcl 4565 . . . . . . 7 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
98simpld 476 . . . . . 6 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
10 prid1g 4431 . . . . . 6 (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷})
119, 10syl 17 . . . . 5 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ {𝐶, 𝐷})
12 eleq2 2839 . . . . 5 ({𝐴} = {𝐶, 𝐷} → (𝐶 ∈ {𝐴} ↔ 𝐶 ∈ {𝐶, 𝐷}))
1311, 12syl5ibrcom 237 . . . 4 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐶 ∈ {𝐴}))
14 elsni 4333 . . . . 5 (𝐶 ∈ {𝐴} → 𝐶 = 𝐴)
1514eqcomd 2777 . . . 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 4537 . . . . . 6 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
198, 18syl 17 . . . . 5 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
2017, 19eleqtrd 2852 . . . 4 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}})
21 elpri 4337 . . . 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 839 . 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 1631  wcel 2145  Vcvv 3351  {csn 4316  {cpr 4318  cop 4322
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323
This theorem is referenced by:  opth  5072  dmsnopg  5748  funcnvsn  6079  oprabid  6822  seqomlem2  7699  unxpdomlem3  8322  dfac5lem4  9149  dcomex  9471  canthwelem  9674  uzrdgfni  12965  gsum2d2  18580  poimirlem9  33751
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