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Theorem opthprneg 4531
Description: An unordered pair has the ordered pair property (compare opth 5072) under certain conditions. Variant of opthpr 4515 in closed form. (Contributed by AV, 13-Jun-2022.)
Assertion
Ref Expression
opthprneg (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))

Proof of Theorem opthprneg
StepHypRef Expression
1 preq12bg 4517 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
21adantlr 694 . . . 4 ((((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
3 idd 24 . . . . . . . 8 (𝐴𝐷 → ((𝐴 = 𝐶𝐵 = 𝐷) → (𝐴 = 𝐶𝐵 = 𝐷)))
4 df-ne 2944 . . . . . . . . . 10 (𝐴𝐷 ↔ ¬ 𝐴 = 𝐷)
5 pm2.21 121 . . . . . . . . . 10 𝐴 = 𝐷 → (𝐴 = 𝐷 → (𝐵 = 𝐶 → (𝐴 = 𝐶𝐵 = 𝐷))))
64, 5sylbi 207 . . . . . . . . 9 (𝐴𝐷 → (𝐴 = 𝐷 → (𝐵 = 𝐶 → (𝐴 = 𝐶𝐵 = 𝐷))))
76impd 396 . . . . . . . 8 (𝐴𝐷 → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐷)))
83, 7jaod 848 . . . . . . 7 (𝐴𝐷 → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → (𝐴 = 𝐶𝐵 = 𝐷)))
9 orc 856 . . . . . . 7 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
108, 9impbid1 215 . . . . . 6 (𝐴𝐷 → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
1110adantl 467 . . . . 5 ((𝐴𝐵𝐴𝐷) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
1211ad2antlr 706 . . . 4 ((((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
132, 12bitrd 268 . . 3 ((((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
1413expcom 398 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷))))
15 ianor 966 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V))
16 simpl 468 . . . . . . . . . . 11 ((𝐴𝐵𝐴𝐷) → 𝐴𝐵)
1716anim2i 603 . . . . . . . . . 10 (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵))
18 df-3an 1073 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑊𝐴𝐵) ↔ ((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵))
1917, 18sylibr 224 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → (𝐴𝑉𝐵𝑊𝐴𝐵))
20 prneprprc 4526 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
2119, 20sylan 569 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
2221ancoms 455 . . . . . . 7 ((¬ 𝐶 ∈ V ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
23 eqneqall 2954 . . . . . . 7 ({𝐴, 𝐵} = {𝐶, 𝐷} → ({𝐴, 𝐵} ≠ {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷)))
2422, 23syl5com 31 . . . . . 6 ((¬ 𝐶 ∈ V ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷)))
25 prneprprc 4526 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐷 ∈ V) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
2619, 25sylan 569 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) ∧ ¬ 𝐷 ∈ V) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
2726ancoms 455 . . . . . . 7 ((¬ 𝐷 ∈ V ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
28 prcom 4403 . . . . . . . . 9 {𝐶, 𝐷} = {𝐷, 𝐶}
2928eqeq2i 2783 . . . . . . . 8 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐷, 𝐶})
30 eqneqall 2954 . . . . . . . 8 ({𝐴, 𝐵} = {𝐷, 𝐶} → ({𝐴, 𝐵} ≠ {𝐷, 𝐶} → (𝐴 = 𝐶𝐵 = 𝐷)))
3129, 30sylbi 207 . . . . . . 7 ({𝐴, 𝐵} = {𝐶, 𝐷} → ({𝐴, 𝐵} ≠ {𝐷, 𝐶} → (𝐴 = 𝐶𝐵 = 𝐷)))
3227, 31syl5com 31 . . . . . 6 ((¬ 𝐷 ∈ V ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷)))
3324, 32jaoian 941 . . . . 5 (((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷)))
34 preq12 4406 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
3533, 34impbid1 215 . . . 4 (((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
3635ex 397 . . 3 ((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) → (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷))))
3715, 36sylbi 207 . 2 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷))))
3814, 37pm2.61i 176 1 (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 382  wo 836  w3a 1071   = wceq 1631  wcel 2145  wne 2943  Vcvv 3351  {cpr 4318
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-3an 1073  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-sn 4317  df-pr 4319
This theorem is referenced by: (None)
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