MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  reusv2lem1 Structured version   Visualization version   GIF version

Theorem reusv2lem1 5398
Description: Lemma for reusv2 5403. (Contributed by NM, 22-Oct-2010.) (Proof shortened by Mario Carneiro, 19-Nov-2016.)
Assertion
Ref Expression
reusv2lem1 (𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝑦𝐴 𝑥 = 𝐵))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵
Allowed substitution hint:   𝐵(𝑦)

Proof of Theorem reusv2lem1
StepHypRef Expression
1 n0 4353 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦𝐴)
2 nfra1 3284 . . . . 5 𝑦𝑦𝐴 𝑥 = 𝐵
32nfmov 2560 . . . 4 𝑦∃*𝑥𝑦𝐴 𝑥 = 𝐵
4 rsp 3247 . . . . . . 7 (∀𝑦𝐴 𝑥 = 𝐵 → (𝑦𝐴𝑥 = 𝐵))
54com12 32 . . . . . 6 (𝑦𝐴 → (∀𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
65alrimiv 1927 . . . . 5 (𝑦𝐴 → ∀𝑥(∀𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
7 mo2icl 3720 . . . . 5 (∀𝑥(∀𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵) → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
86, 7syl 17 . . . 4 (𝑦𝐴 → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
93, 8exlimi 2217 . . 3 (∃𝑦 𝑦𝐴 → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
101, 9sylbi 217 . 2 (𝐴 ≠ ∅ → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
11 df-eu 2569 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ (∃𝑥𝑦𝐴 𝑥 = 𝐵 ∧ ∃*𝑥𝑦𝐴 𝑥 = 𝐵))
1211rbaib 538 . 2 (∃*𝑥𝑦𝐴 𝑥 = 𝐵 → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝑦𝐴 𝑥 = 𝐵))
1310, 12syl 17 1 (𝐴 ≠ ∅ → (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ ∃𝑥𝑦𝐴 𝑥 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1538   = wceq 1540  wex 1779  wcel 2108  ∃*wmo 2538  ∃!weu 2568  wne 2940  wral 3061  c0 4333
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-v 3482  df-dif 3954  df-nul 4334
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator