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Theorem opth1 3947
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 3505 . . 3 ({A} = {𝐶} → A = 𝐶)
32a1i 9 . 2 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ({A} = {𝐶} → A = 𝐶))
4 opth1.2 . . . . . . . . 9 B V
51, 4opi1 3943 . . . . . . . 8 {A} A, B
6 id 19 . . . . . . . 8 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ⟨A, B⟩ = ⟨𝐶, 𝐷⟩)
75, 6syl5eleq 2108 . . . . . . 7 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {A} 𝐶, 𝐷⟩)
8 oprcl 3547 . . . . . . 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 3448 . . . . 5 (𝐶 V → 𝐶 {𝐶, 𝐷})
1210, 11syl 14 . . . 4 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → 𝐶 {𝐶, 𝐷})
13 eleq2 2083 . . . 4 ({A} = {𝐶, 𝐷} → (𝐶 {A} ↔ 𝐶 {𝐶, 𝐷}))
1412, 13syl5ibrcom 146 . . 3 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ({A} = {𝐶, 𝐷} → 𝐶 {A}))
15 elsni 3374 . . . 4 (𝐶 {A} → 𝐶 = A)
1615eqcomd 2027 . . 3 (𝐶 {A} → A = 𝐶)
1714, 16syl6 29 . 2 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → ({A} = {𝐶, 𝐷} → A = 𝐶))
18 dfopg 3521 . . . . 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 2098 . . 3 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → {A} {{𝐶}, {𝐶, 𝐷}})
21 elpri 3370 . . 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 625 1 (⟨A, B⟩ = ⟨𝐶, 𝐷⟩ → A = 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   wo 616   = wceq 1228   wcel 1374  Vcvv 2535  {csn 3350  {cpr 3351  cop 3353
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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901
This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359
This theorem is referenced by:  opth  3948  dmsnopg  4719  funcnvsn  4871  oprabid  5461
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