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Theorem opth1 3964
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
21sneqr 3522 . . 3 ({A} = {𝐶} → A = 𝐶)
32a1i 9 . 2 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ({A} = {𝐶} → A = 𝐶))
4 opth1.2 . . . . . . . . 9 B V
51, 4opi1 3960 . . . . . . . 8 {A} A, B
6 id 19 . . . . . . . 8 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨A, B⟩ = ⟨𝐶, 𝐷⟩)
75, 6syl5eleq 2123 . . . . . . 7 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {A} 𝐶, 𝐷⟩)
8 oprcl 3564 . . . . . . 7 ({A} 𝐶, 𝐷⟩ → (𝐶 V 𝐷 V))
97, 8syl 14 . . . . . 6 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → (𝐶 V 𝐷 V))
109simpld 105 . . . . 5 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 V)
11 prid1g 3465 . . . . 5 (𝐶 V → 𝐶 {𝐶, 𝐷})
1210, 11syl 14 . . . 4 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 {𝐶, 𝐷})
13 eleq2 2098 . . . 4 ({A} = {𝐶, 𝐷} → (𝐶 {A} ↔ 𝐶 {𝐶, 𝐷}))
1412, 13syl5ibrcom 146 . . 3 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ({A} = {𝐶, 𝐷} → 𝐶 {A}))
15 elsni 3391 . . . 4 (𝐶 {A} → 𝐶 = A)
1615eqcomd 2042 . . 3 (𝐶 {A} → A = 𝐶)
1714, 16syl6 29 . 2 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ({A} = {𝐶, 𝐷} → A = 𝐶))
18 dfopg 3538 . . . . 5 ((𝐶 V 𝐷 V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
197, 8, 183syl 17 . . . 4 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
207, 19eleqtrd 2113 . . 3 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {A} {{𝐶}, {𝐶, 𝐷}})
21 elpri 3387 . . 3 ({A} {{𝐶}, {𝐶, 𝐷}} → ({A} = {𝐶} {A} = {𝐶, 𝐷}))
2220, 21syl 14 . 2 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ({A} = {𝐶} {A} = {𝐶, 𝐷}))
233, 17, 22mpjaod 637 1 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → A = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 628   = wceq 1242   wcel 1390  Vcvv 2551  {csn 3367  {cpr 3368  cop 3370
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376
This theorem is referenced by:  opth  3965  dmsnopg  4735  funcnvsn  4888  oprabid  5480
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