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Theorem preq12nebg 4751
 Description: Equality relationship for two proper unordered pairs. (Contributed by AV, 12-Jun-2022.)
Assertion
Ref Expression
preq12nebg ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))

Proof of Theorem preq12nebg
StepHypRef Expression
1 3simpa 1146 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐴𝑉𝐵𝑊))
21anim1i 618 . . . . 5 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ((𝐴𝑉𝐵𝑊) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)))
32ancoms 463 . . . 4 (((𝐶 ∈ V ∧ 𝐷 ∈ V) ∧ (𝐴𝑉𝐵𝑊𝐴𝐵)) → ((𝐴𝑉𝐵𝑊) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)))
4 preq12bg 4742 . . . 4 (((𝐴𝑉𝐵𝑊) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
53, 4syl 17 . . 3 (((𝐶 ∈ V ∧ 𝐷 ∈ V) ∧ (𝐴𝑉𝐵𝑊𝐴𝐵)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
65ex 417 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
7 ianor 980 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V))
8 prneprprc 4749 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
98ancoms 463 . . . . . . 7 ((¬ 𝐶 ∈ V ∧ (𝐴𝑉𝐵𝑊𝐴𝐵)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
10 eqneqall 2963 . . . . . . 7 ({𝐴, 𝐵} = {𝐶, 𝐷} → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
119, 10syl5com 31 . . . . . 6 ((¬ 𝐶 ∈ V ∧ (𝐴𝑉𝐵𝑊𝐴𝐵)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
12 prneprprc 4749 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐷 ∈ V) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
1312ancoms 463 . . . . . . 7 ((¬ 𝐷 ∈ V ∧ (𝐴𝑉𝐵𝑊𝐴𝐵)) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
14 prcom 4626 . . . . . . . . 9 {𝐶, 𝐷} = {𝐷, 𝐶}
1514eqeq2i 2772 . . . . . . . 8 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐷, 𝐶})
16 eqneqall 2963 . . . . . . . 8 ({𝐴, 𝐵} = {𝐷, 𝐶} → ({𝐴, 𝐵} ≠ {𝐷, 𝐶} → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
1715, 16sylbi 220 . . . . . . 7 ({𝐴, 𝐵} = {𝐶, 𝐷} → ({𝐴, 𝐵} ≠ {𝐷, 𝐶} → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
1813, 17syl5com 31 . . . . . 6 ((¬ 𝐷 ∈ V ∧ (𝐴𝑉𝐵𝑊𝐴𝐵)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
1911, 18jaoian 955 . . . . 5 (((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) ∧ (𝐴𝑉𝐵𝑊𝐴𝐵)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
20 preq12 4629 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
21 preq12 4629 . . . . . . 7 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
22 prcom 4626 . . . . . . 7 {𝐷, 𝐶} = {𝐶, 𝐷}
2321, 22eqtrdi 2810 . . . . . 6 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
2420, 23jaoi 855 . . . . 5 (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → {𝐴, 𝐵} = {𝐶, 𝐷})
2519, 24impbid1 228 . . . 4 (((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) ∧ (𝐴𝑉𝐵𝑊𝐴𝐵)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
2625ex 417 . . 3 ((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
277, 26sylbi 220 . 2 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
286, 27pm2.61i 185 1 ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∨ wo 845   ∧ w3a 1085   = wceq 1539   ∈ wcel 2112   ≠ wne 2952  Vcvv 3410  {cpr 4525 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2730 This theorem depends on definitions:  df-bi 210  df-an 401  df-or 846  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2071  df-clab 2737  df-cleq 2751  df-clel 2831  df-nfc 2902  df-ne 2953  df-v 3412  df-dif 3862  df-un 3864  df-nul 4227  df-sn 4524  df-pr 4526 This theorem is referenced by:  prel12g  4752  opthhausdorff  5377  preleqg  9104  pr2cv  40613
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