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Theorem preq12nebg 4583
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 1179 . . . . . 6 ((𝐴𝑉𝐵𝑊𝐴𝐵) → (𝐴𝑉𝐵𝑊))
21anim1i 609 . . . . 5 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ((𝐴𝑉𝐵𝑊) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)))
32ancoms 451 . . . 4 (((𝐶 ∈ V ∧ 𝐷 ∈ V) ∧ (𝐴𝑉𝐵𝑊𝐴𝐵)) → ((𝐴𝑉𝐵𝑊) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)))
4 preq12bg 4571 . . . 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 402 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
7 ianor 1005 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V))
8 prneprprc 4580 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
98ancoms 451 . . . . . . 7 ((¬ 𝐶 ∈ V ∧ (𝐴𝑉𝐵𝑊𝐴𝐵)) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
10 eqneqall 2982 . . . . . . 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 4580 . . . . . . . 8 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐷 ∈ V) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
1312ancoms 451 . . . . . . 7 ((¬ 𝐷 ∈ V ∧ (𝐴𝑉𝐵𝑊𝐴𝐵)) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
14 prcom 4456 . . . . . . . . 9 {𝐶, 𝐷} = {𝐷, 𝐶}
1514eqeq2i 2811 . . . . . . . 8 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐷, 𝐶})
16 eqneqall 2982 . . . . . . . 8 ({𝐴, 𝐵} = {𝐷, 𝐶} → ({𝐴, 𝐵} ≠ {𝐷, 𝐶} → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
1715, 16sylbi 209 . . . . . . 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 980 . . . . 5 (((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) ∧ (𝐴𝑉𝐵𝑊𝐴𝐵)) → ({𝐴, 𝐵} = {𝐶, 𝐷} → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
20 preq12 4459 . . . . . 6 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
21 preq12 4459 . . . . . . 7 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐷, 𝐶})
22 prcom 4456 . . . . . . 7 {𝐷, 𝐶} = {𝐶, 𝐷}
2321, 22syl6eq 2849 . . . . . 6 ((𝐴 = 𝐷𝐵 = 𝐶) → {𝐴, 𝐵} = {𝐶, 𝐷})
2420, 23jaoi 884 . . . . 5 (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → {𝐴, 𝐵} = {𝐶, 𝐷})
2519, 24impbid1 217 . . . 4 (((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) ∧ (𝐴𝑉𝐵𝑊𝐴𝐵)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
2625ex 402 . . 3 ((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
277, 26sylbi 209 . 2 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → ((𝐴𝑉𝐵𝑊𝐴𝐵) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))))
286, 27pm2.61i 177 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 198  wa 385  wo 874  w3a 1108   = wceq 1653  wcel 2157  wne 2971  Vcvv 3385  {cpr 4370
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ne 2972  df-v 3387  df-dif 3772  df-un 3774  df-nul 4116  df-sn 4369  df-pr 4371
This theorem is referenced by:  prel12g  4584  opthhausdorff  5173  preleqg  8760
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