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

Theorem elpreqprlem 4788
Description: Lemma for elpreqpr 4789. (Contributed by Scott Fenton, 7-Dec-2020.) (Revised by AV, 9-Dec-2020.)
Assertion
Ref Expression
elpreqprlem (𝐵𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})
Distinct variable groups:   𝑥,𝐵   𝑥,𝐶
Allowed substitution hint:   𝑉(𝑥)

Proof of Theorem elpreqprlem
StepHypRef Expression
1 eqid 2818 . . . 4 {𝐵, 𝐶} = {𝐵, 𝐶}
2 preq2 4662 . . . . . 6 (𝑥 = 𝐶 → {𝐵, 𝑥} = {𝐵, 𝐶})
32eqeq2d 2829 . . . . 5 (𝑥 = 𝐶 → ({𝐵, 𝐶} = {𝐵, 𝑥} ↔ {𝐵, 𝐶} = {𝐵, 𝐶}))
43spcegv 3594 . . . 4 (𝐶 ∈ V → ({𝐵, 𝐶} = {𝐵, 𝐶} → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
51, 4mpi 20 . . 3 (𝐶 ∈ V → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})
65a1d 25 . 2 (𝐶 ∈ V → (𝐵𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
7 dfsn2 4570 . . . 4 {𝐵} = {𝐵, 𝐵}
8 preq2 4662 . . . . . 6 (𝑥 = 𝐵 → {𝐵, 𝑥} = {𝐵, 𝐵})
98eqeq2d 2829 . . . . 5 (𝑥 = 𝐵 → ({𝐵} = {𝐵, 𝑥} ↔ {𝐵} = {𝐵, 𝐵}))
109spcegv 3594 . . . 4 (𝐵𝑉 → ({𝐵} = {𝐵, 𝐵} → ∃𝑥{𝐵} = {𝐵, 𝑥}))
117, 10mpi 20 . . 3 (𝐵𝑉 → ∃𝑥{𝐵} = {𝐵, 𝑥})
12 prprc2 4694 . . . . 5 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵})
1312eqeq1d 2820 . . . 4 𝐶 ∈ V → ({𝐵, 𝐶} = {𝐵, 𝑥} ↔ {𝐵} = {𝐵, 𝑥}))
1413exbidv 1913 . . 3 𝐶 ∈ V → (∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥} ↔ ∃𝑥{𝐵} = {𝐵, 𝑥}))
1511, 14syl5ibr 247 . 2 𝐶 ∈ V → (𝐵𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥}))
166, 15pm2.61i 183 1 (𝐵𝑉 → ∃𝑥{𝐵, 𝐶} = {𝐵, 𝑥})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1528  wex 1771  wcel 2105  Vcvv 3492  {csn 4557  {cpr 4559
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-dif 3936  df-un 3938  df-nul 4289  df-sn 4558  df-pr 4560
This theorem is referenced by:  elpreqpr  4789
  Copyright terms: Public domain W3C validator