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

Theorem elpreqpr 4770
Description: Equality and membership rule for pairs. (Contributed by Scott Fenton, 7-Dec-2020.)
Assertion
Ref Expression
elpreqpr (𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem elpreqpr
StepHypRef Expression
1 elpri 4562 . 2 (𝐴 ∈ {𝐵, 𝐶} → (𝐴 = 𝐵𝐴 = 𝐶))
2 elex 3489 . 2 (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V)
3 elpreqprlem 4769 . . . . 5 (𝐵 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})
4 eleq1 2899 . . . . . 6 (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
5 preq1 4642 . . . . . . . 8 (𝐴 = 𝐵 → {𝐴, 𝑥} = {𝐵, 𝑥})
65eqeq2d 2832 . . . . . . 7 (𝐴 = 𝐵 → ({𝐵, 𝐶} = {𝐴, 𝑥} ↔ {𝐵, 𝐶} = {𝐵, 𝑥}))
76exbidv 1923 . . . . . 6 (𝐴 = 𝐵 → (∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
84, 7imbi12d 348 . . . . 5 (𝐴 = 𝐵 → ((𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}) ↔ (𝐵 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})))
93, 8mpbiri 261 . . . 4 (𝐴 = 𝐵 → (𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}))
109imp 410 . . 3 ((𝐴 = 𝐵𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
11 elpreqprlem 4769 . . . . . 6 (𝐶 ∈ V → ∃𝑥{𝐶, 𝐵} = {𝐶, 𝑥})
12 prcom 4641 . . . . . . . 8 {𝐶, 𝐵} = {𝐵, 𝐶}
1312eqeq1i 2826 . . . . . . 7 ({𝐶, 𝐵} = {𝐶, 𝑥} ↔ {𝐵, 𝐶} = {𝐶, 𝑥})
1413exbii 1849 . . . . . 6 (∃𝑥{𝐶, 𝐵} = {𝐶, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})
1511, 14sylib 221 . . . . 5 (𝐶 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})
16 eleq1 2899 . . . . . 6 (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
17 preq1 4642 . . . . . . . 8 (𝐴 = 𝐶 → {𝐴, 𝑥} = {𝐶, 𝑥})
1817eqeq2d 2832 . . . . . . 7 (𝐴 = 𝐶 → ({𝐵, 𝐶} = {𝐴, 𝑥} ↔ {𝐵, 𝐶} = {𝐶, 𝑥}))
1918exbidv 1923 . . . . . 6 (𝐴 = 𝐶 → (∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥} ↔ ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥}))
2016, 19imbi12d 348 . . . . 5 (𝐴 = 𝐶 → ((𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}) ↔ (𝐶 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐶, 𝑥})))
2115, 20mpbiri 261 . . . 4 (𝐴 = 𝐶 → (𝐴 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥}))
2221imp 410 . . 3 ((𝐴 = 𝐶𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
2310, 22jaoian 954 . 2 (((𝐴 = 𝐵𝐴 = 𝐶) ∧ 𝐴 ∈ V) → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
241, 2, 23syl2anc 587 1 (𝐴 ∈ {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐴, 𝑥})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wo 844   = wceq 1538  wex 1781  wcel 2115  Vcvv 3471  {cpr 4542
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-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  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-nul 4267  df-sn 4541  df-pr 4543
This theorem is referenced by:  elpreqprb  4771
  Copyright terms: Public domain W3C validator