MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eupth2lem1 Structured version   Visualization version   GIF version

Theorem eupth2lem1 26938
Description: Lemma for eupth2 26959. (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 2693 . . 3 (∅ = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ ∅ ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
21bibi1d 333 . 2 (∅ = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → ((𝑈 ∈ ∅ ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))) ↔ (𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)))))
3 eleq2 2693 . . 3 ({𝐴, 𝐵} = if(𝐴 = 𝐵, ∅, {𝐴, 𝐵}) → (𝑈 ∈ {𝐴, 𝐵} ↔ 𝑈 ∈ if(𝐴 = 𝐵, ∅, {𝐴, 𝐵})))
43bibi1d 333 . 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 3900 . . . 4 ¬ 𝑈 ∈ ∅
65a1i 11 . . 3 ((𝑈𝑉𝐴 = 𝐵) → ¬ 𝑈 ∈ ∅)
7 simpl 473 . . . . 5 ((𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)) → 𝐴𝐵)
87neneqd 2801 . . . 4 ((𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)) → ¬ 𝐴 = 𝐵)
9 simpr 477 . . . 4 ((𝑈𝑉𝐴 = 𝐵) → 𝐴 = 𝐵)
108, 9nsyl3 133 . . 3 ((𝑈𝑉𝐴 = 𝐵) → ¬ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵)))
116, 102falsed 366 . 2 ((𝑈𝑉𝐴 = 𝐵) → (𝑈 ∈ ∅ ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
12 elprg 4172 . . 3 (𝑈𝑉 → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝑈 = 𝐴𝑈 = 𝐵)))
13 df-ne 2797 . . . 4 (𝐴𝐵 ↔ ¬ 𝐴 = 𝐵)
14 ibar 525 . . . 4 (𝐴𝐵 → ((𝑈 = 𝐴𝑈 = 𝐵) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
1513, 14sylbir 225 . . 3 𝐴 = 𝐵 → ((𝑈 = 𝐴𝑈 = 𝐵) ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
1612, 15sylan9bb 735 . 2 ((𝑈𝑉 ∧ ¬ 𝐴 = 𝐵) → (𝑈 ∈ {𝐴, 𝐵} ↔ (𝐴𝐵 ∧ (𝑈 = 𝐴𝑈 = 𝐵))))
172, 4, 11, 16ifbothda 4100 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  wo 383  wa 384   = wceq 1480  wcel 1992  wne 2796  c0 3896  ifcif 4063  {cpr 4155
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-v 3193  df-dif 3563  df-un 3565  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156
This theorem is referenced by:  eupth2lem2  26939  eupth2lem3lem6  26953
  Copyright terms: Public domain W3C validator