MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  opth1 Structured version   Visualization version   GIF version

Theorem opth1 4934
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 4928 . . 3 {𝐴} ∈ ⟨𝐴, 𝐵
4 id 22 . . 3 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → ⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩)
53, 4syl5eleq 2705 . 2 (⟨𝐴, 𝐵⟩ = ⟨𝐶, 𝐷⟩ → {𝐴} ∈ ⟨𝐶, 𝐷⟩)
61sneqr 4362 . . . 4 ({𝐴} = {𝐶} → 𝐴 = 𝐶)
76a1i 11 . . 3 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶} → 𝐴 = 𝐶))
8 oprcl 4418 . . . . . . 7 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → (𝐶 ∈ V ∧ 𝐷 ∈ V))
98simpld 475 . . . . . 6 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ V)
10 prid1g 4286 . . . . . 6 (𝐶 ∈ V → 𝐶 ∈ {𝐶, 𝐷})
119, 10syl 17 . . . . 5 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → 𝐶 ∈ {𝐶, 𝐷})
12 eleq2 2688 . . . . 5 ({𝐴} = {𝐶, 𝐷} → (𝐶 ∈ {𝐴} ↔ 𝐶 ∈ {𝐶, 𝐷}))
1311, 12syl5ibrcom 237 . . . 4 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ({𝐴} = {𝐶, 𝐷} → 𝐶 ∈ {𝐴}))
14 elsni 4185 . . . . 5 (𝐶 ∈ {𝐴} → 𝐶 = 𝐴)
1514eqcomd 2626 . . . 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 4391 . . . . . 6 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
198, 18syl 17 . . . . 5 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → ⟨𝐶, 𝐷⟩ = {{𝐶}, {𝐶, 𝐷}})
2017, 19eleqtrd 2701 . . . 4 ({𝐴} ∈ ⟨𝐶, 𝐷⟩ → {𝐴} ∈ {{𝐶}, {𝐶, 𝐷}})
21 elpri 4188 . . . 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 396 . 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  wo 383  wa 384   = wceq 1481  wcel 1988  Vcvv 3195  {csn 4168  {cpr 4170  cop 4174
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-v 3197  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175
This theorem is referenced by:  opth  4935  dmsnopg  5594  funcnvsn  5924  oprabid  6662  seqomlem2  7531  unxpdomlem3  8151  dfac5lem4  8934  dcomex  9254  canthwelem  9457  uzrdgfni  12740  gsum2d2  18354  poimirlem9  33389
  Copyright terms: Public domain W3C validator