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Theorem opth1 5340
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 5333 . . 3 {𝐴} ∈ ⟨𝐴, 𝐵
4 id 22 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
53, 4eleqtrid 2918 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
61sneqr 4744 . . . 4 ({𝐴} = {𝐶} → 𝐴 = 𝐶)
76a1i 11 . . 3 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶} → 𝐴 = 𝐶))
8 oprcl 4802 . . . . . . 7 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
98simpld 498 . . . . . 6 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
10 prid1g 4669 . . . . . 6 (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷})
119, 10syl 17 . . . . 5 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ {𝐶, 𝐷})
12 eleq2 2900 . . . . 5 ({𝐴} = {𝐶, 𝐷} → (𝐶 ∈ {𝐴} ↔ 𝐶 ∈ {𝐶, 𝐷}))
1311, 12syl5ibrcom 250 . . . 4 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐶 ∈ {𝐴}))
14 elsni 4557 . . . . 5 (𝐶 ∈ {𝐴} → 𝐶 = 𝐴)
1514eqcomd 2827 . . . 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 4774 . . . . . 6 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
198, 18syl 17 . . . . 5 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
2017, 19eleqtrd 2914 . . . 4 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}})
21 elpri 4562 . . . 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 857 . 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 399  wo 844   = wceq 1538  wcel 2115  Vcvv 3471  {csn 4540  {cpr 4542  cop 4546
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793  ax-sep 5176  ax-nul 5183  ax-pr 5303
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547
This theorem is referenced by:  opth  5341  dmsnopg  6043  funcnvsn  6377  oprabidw  7161  oprabid  7162  seqomlem2  8062  unxpdomlem3  8700  dfac5lem4  9529  dcomex  9846  canthwelem  10049  uzrdgfni  13309  fnpr2ob  16810  gsum2d2  19073  poimirlem9  34952  ichnreuop  43812  ichreuopeq  43813
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