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Theorem eupth2lem1 30100
Description: Lemma for eupth2 30121. (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 2814 . . 3 (∅ = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ ∅ ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
21bibi1d 342 . 2 (∅ = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → ((𝑈 ∈ ∅ ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))) ↔ (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)))))
3 eleq2 2814 . . 3 ({𝐴, 𝐵} = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ {𝐴, 𝐵} ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
43bibi1d 342 . 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 4330 . . . 4 ¬ 𝑈 ∈ ∅
65a1i 11 . . 3 ((𝑈𝑉𝐴 = 𝐵) → ¬ 𝑈 ∈ ∅)
7 simpl 481 . . . . 5 ((𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)) → 𝐴𝐵)
87neneqd 2934 . . . 4 ((𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)) → ¬ 𝐴 = 𝐵)
9 simpr 483 . . . 4 ((𝑈𝑉𝐴 = 𝐵) → 𝐴 = 𝐵)
108, 9nsyl3 138 . . 3 ((𝑈𝑉𝐴 = 𝐵) → ¬ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)))
116, 102falsed 375 . 2 ((𝑈𝑉𝐴 = 𝐵) → (𝑈 ∈ ∅ ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
12 elprg 4652 . . 3 (𝑈𝑉 → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝑈 = 𝐴𝑈 = 𝐵)))
13 df-ne 2930 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
14 ibar 527 . . . 4 (𝐴𝐵 → ((𝑈 = 𝐴𝑈 = 𝐵) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
1513, 14sylbir 234 . . 3 𝐴 = 𝐵 → ((𝑈 = 𝐴𝑈 = 𝐵) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
1612, 15sylan9bb 508 . 2 ((𝑈𝑉 ∧ ¬ 𝐴 = 𝐵) → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
172, 4, 11, 16ifbothda 4568 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 205  wa 394  wo 845   = wceq 1533  wcel 2098  wne 2929  c0 4322  ifcif 4530  {cpr 4632
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2696
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2703  df-cleq 2717  df-clel 2802  df-ne 2930  df-v 3463  df-dif 3947  df-un 3949  df-nul 4323  df-if 4531  df-sn 4631  df-pr 4633
This theorem is referenced by:  eupth2lem2  30101  eupth2lem3lem6  30115
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