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Theorem opthprneg 4586
Description: An unordered pair has the ordered pair property (compare opth 5136) under certain conditions. Variant of opthpr 4570 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 4572 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
21adantlr 707 . . . 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 2973 . . . . . . . . . 10 (𝐴𝐷 ↔ ¬ 𝐴 = 𝐷)
5 pm2.21 121 . . . . . . . . . 10 𝐴 = 𝐷 → (𝐴 = 𝐷 → (𝐵 = 𝐶 → (𝐴 = 𝐶𝐵 = 𝐷))))
64, 5sylbi 209 . . . . . . . . 9 (𝐴𝐷 → (𝐴 = 𝐷 → (𝐵 = 𝐶 → (𝐴 = 𝐶𝐵 = 𝐷))))
76impd 399 . . . . . . . 8 (𝐴𝐷 → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐷)))
83, 7jaod 886 . . . . . . 7 (𝐴𝐷 → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → (𝐴 = 𝐶𝐵 = 𝐷)))
9 orc 894 . . . . . . 7 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
108, 9impbid1 217 . . . . . 6 (𝐴𝐷 → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
1110adantl 474 . . . . 5 ((𝐴𝐵𝐴𝐷) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
1211ad2antlr 719 . . . 4 ((((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
132, 12bitrd 271 . . 3 ((((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
1413expcom 403 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷))))
15 ianor 1005 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V))
16 simpl 475 . . . . . . . . . . 11 ((𝐴𝐵𝐴𝐷) → 𝐴𝐵)
1716anim2i 611 . . . . . . . . . 10 (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵))
18 df-3an 1110 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑊𝐴𝐵) ↔ ((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵))
1917, 18sylibr 226 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → (𝐴𝑉𝐵𝑊𝐴𝐵))
20 prneprprc 4581 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
2119, 20sylan 576 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
2221ancoms 451 . . . . . . 7 ((¬ 𝐶 ∈ V ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
23 eqneqall 2983 . . . . . . 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 4581 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐷 ∈ V) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
2619, 25sylan 576 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) ∧ ¬ 𝐷 ∈ V) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
2726ancoms 451 . . . . . . 7 ((¬ 𝐷 ∈ V ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
28 prcom 4457 . . . . . . . . 9 {𝐶, 𝐷} = {𝐷, 𝐶}
2928eqeq2i 2812 . . . . . . . 8 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐷, 𝐶})
30 eqneqall 2983 . . . . . . . 8 ({𝐴, 𝐵} = {𝐷, 𝐶} → ({𝐴, 𝐵} ≠ {𝐷, 𝐶} → (𝐴 = 𝐶𝐵 = 𝐷)))
3129, 30sylbi 209 . . . . . . 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 980 . . . . 5 (((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷)))
34 preq12 4460 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
3533, 34impbid1 217 . . . 4 (((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
3635ex 402 . . 3 ((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) → (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷))))
3715, 36sylbi 209 . 2 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷))))
3814, 37pm2.61i 177 1 (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
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 2972  Vcvv 3386  {cpr 4371
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 2378  ax-ext 2778
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 2787  df-cleq 2793  df-clel 2796  df-nfc 2931  df-ne 2973  df-v 3388  df-dif 3773  df-un 3775  df-nul 4117  df-sn 4370  df-pr 4372
This theorem is referenced by: (None)
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