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Theorem reusv2lem1 5017
Description: Lemma for reusv2 5023. (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 4074 . . 3 (𝐴 ≠ ∅ ↔ ∃𝑦 𝑦𝐴)
2 nfra1 3079 . . . . 5 𝑦𝑦𝐴 𝑥 = 𝐵
32nfmo 2624 . . . 4 𝑦∃*𝑥𝑦𝐴 𝑥 = 𝐵
4 rsp 3067 . . . . . . 7 (∀𝑦𝐴 𝑥 = 𝐵 → (𝑦𝐴𝑥 = 𝐵))
54com12 32 . . . . . 6 (𝑦𝐴 → (∀𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
65alrimiv 2004 . . . . 5 (𝑦𝐴 → ∀𝑥(∀𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵))
7 mo2icl 3526 . . . . 5 (∀𝑥(∀𝑦𝐴 𝑥 = 𝐵𝑥 = 𝐵) → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
86, 7syl 17 . . . 4 (𝑦𝐴 → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
93, 8exlimi 2233 . . 3 (∃𝑦 𝑦𝐴 → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
101, 9sylbi 207 . 2 (𝐴 ≠ ∅ → ∃*𝑥𝑦𝐴 𝑥 = 𝐵)
11 eu5 2633 . . 3 (∃!𝑥𝑦𝐴 𝑥 = 𝐵 ↔ (∃𝑥𝑦𝐴 𝑥 = 𝐵 ∧ ∃*𝑥𝑦𝐴 𝑥 = 𝐵))
1211rbaib 985 . 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 196  wal 1630   = wceq 1632  wex 1853  wcel 2139  ∃!weu 2607  ∃*wmo 2608  wne 2932  wral 3050  c0 4058
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-v 3342  df-dif 3718  df-nul 4059
This theorem is referenced by: (None)
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