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Theorem eupth2lem1 27398
Description: Lemma for eupth2 27419. (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 2839 . . 3 (∅ = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ ∅ ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
21bibi1d 332 . 2 (∅ = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → ((𝑈 ∈ ∅ ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))) ↔ (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)))))
3 eleq2 2839 . . 3 ({𝐴, 𝐵} = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ {𝐴, 𝐵} ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
43bibi1d 332 . 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 4067 . . . 4 ¬ 𝑈 ∈ ∅
65a1i 11 . . 3 ((𝑈𝑉𝐴 = 𝐵) → ¬ 𝑈 ∈ ∅)
7 simpl 468 . . . . 5 ((𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)) → 𝐴𝐵)
87neneqd 2948 . . . 4 ((𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)) → ¬ 𝐴 = 𝐵)
9 simpr 471 . . . 4 ((𝑈𝑉𝐴 = 𝐵) → 𝐴 = 𝐵)
108, 9nsyl3 135 . . 3 ((𝑈𝑉𝐴 = 𝐵) → ¬ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)))
116, 102falsed 365 . 2 ((𝑈𝑉𝐴 = 𝐵) → (𝑈 ∈ ∅ ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
12 elprg 4337 . . 3 (𝑈𝑉 → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝑈 = 𝐴𝑈 = 𝐵)))
13 df-ne 2944 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
14 ibar 518 . . . 4 (𝐴𝐵 → ((𝑈 = 𝐴𝑈 = 𝐵) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
1513, 14sylbir 225 . . 3 𝐴 = 𝐵 → ((𝑈 = 𝐴𝑈 = 𝐵) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
1612, 15sylan9bb 499 . 2 ((𝑈𝑉 ∧ ¬ 𝐴 = 𝐵) → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
172, 4, 11, 16ifbothda 4263 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 196  wa 382  wo 836   = wceq 1631  wcel 2145  wne 2943  c0 4063  ifcif 4226  {cpr 4319
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-v 3353  df-dif 3726  df-un 3728  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320
This theorem is referenced by:  eupth2lem2  27399  eupth2lem3lem6  27413
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