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Theorem eupth2lem1 27924
Description: Lemma for eupth2 27945. (Contributed by Mario Carneiro, 8-Apr-2015.)
Assertion
Ref Expression
eupth2lem1 (𝑈𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))

Proof of Theorem eupth2lem1
StepHypRef Expression
1 eleq2 2898 . . 3 (∅ = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ ∅ ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
21bibi1d 345 . 2 (∅ = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → ((𝑈 ∈ ∅ ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))) ↔ (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)))))
3 eleq2 2898 . . 3 ({𝐴, 𝐵} = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ {𝐴, 𝐵} ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
43bibi1d 345 . 2 ({𝐴, 𝐵} = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → ((𝑈 ∈ {𝐴, 𝐵} ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))) ↔ (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)))))
5 noel 4293 . . . 4 ¬ 𝑈 ∈ ∅
65a1i 11 . . 3 ((𝑈𝑉𝐴 = 𝐵) → ¬ 𝑈 ∈ ∅)
7 simpl 483 . . . . 5 ((𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)) → 𝐴𝐵)
87neneqd 3018 . . . 4 ((𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)) → ¬ 𝐴 = 𝐵)
9 simpr 485 . . . 4 ((𝑈𝑉𝐴 = 𝐵) → 𝐴 = 𝐵)
108, 9nsyl3 140 . . 3 ((𝑈𝑉𝐴 = 𝐵) → ¬ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)))
116, 102falsed 378 . 2 ((𝑈𝑉𝐴 = 𝐵) → (𝑈 ∈ ∅ ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
12 elprg 4578 . . 3 (𝑈𝑉 → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝑈 = 𝐴𝑈 = 𝐵)))
13 df-ne 3014 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
14 ibar 529 . . . 4 (𝐴𝐵 → ((𝑈 = 𝐴𝑈 = 𝐵) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
1513, 14sylbir 236 . . 3 𝐴 = 𝐵 → ((𝑈 = 𝐴𝑈 = 𝐵) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
1612, 15sylan9bb 510 . 2 ((𝑈𝑉 ∧ ¬ 𝐴 = 𝐵) → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
172, 4, 11, 16ifbothda 4500 1 (𝑈𝑉 → (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841   = wceq 1528  wcel 2105  wne 3013  c0 4288  ifcif 4463  {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-ne 3014  df-v 3494  df-dif 3936  df-un 3938  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560
This theorem is referenced by:  eupth2lem2  27925  eupth2lem3lem6  27939
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