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

Theorem opthprneg 4793
Description: An unordered pair has the ordered pair property (compare opth 5364) under certain conditions. Variant of opthpr 4780 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 4782 . . . . 5 (((𝐴𝑉𝐵𝑊) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶))))
21adantlr 711 . . . 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 3021 . . . . . . . . . 10 (𝐴𝐷 ↔ ¬ 𝐴 = 𝐷)
5 pm2.21 123 . . . . . . . . . 10 𝐴 = 𝐷 → (𝐴 = 𝐷 → (𝐵 = 𝐶 → (𝐴 = 𝐶𝐵 = 𝐷))))
64, 5sylbi 218 . . . . . . . . 9 (𝐴𝐷 → (𝐴 = 𝐷 → (𝐵 = 𝐶 → (𝐴 = 𝐶𝐵 = 𝐷))))
76impd 411 . . . . . . . 8 (𝐴𝐷 → ((𝐴 = 𝐷𝐵 = 𝐶) → (𝐴 = 𝐶𝐵 = 𝐷)))
83, 7jaod 855 . . . . . . 7 (𝐴𝐷 → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) → (𝐴 = 𝐶𝐵 = 𝐷)))
9 orc 863 . . . . . . 7 ((𝐴 = 𝐶𝐵 = 𝐷) → ((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)))
108, 9impbid1 226 . . . . . 6 (𝐴𝐷 → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
1110adantl 482 . . . . 5 ((𝐴𝐵𝐴𝐷) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
1211ad2antlr 723 . . . 4 ((((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → (((𝐴 = 𝐶𝐵 = 𝐷) ∨ (𝐴 = 𝐷𝐵 = 𝐶)) ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
132, 12bitrd 280 . . 3 ((((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) ∧ (𝐶 ∈ V ∧ 𝐷 ∈ V)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
1413expcom 414 . 2 ((𝐶 ∈ V ∧ 𝐷 ∈ V) → (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷))))
15 ianor 977 . . 3 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V))
16 simpl 483 . . . . . . . . . . 11 ((𝐴𝐵𝐴𝐷) → 𝐴𝐵)
1716anim2i 616 . . . . . . . . . 10 (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵))
18 df-3an 1083 . . . . . . . . . 10 ((𝐴𝑉𝐵𝑊𝐴𝐵) ↔ ((𝐴𝑉𝐵𝑊) ∧ 𝐴𝐵))
1917, 18sylibr 235 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → (𝐴𝑉𝐵𝑊𝐴𝐵))
20 prneprprc 4789 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
2119, 20sylan 580 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) ∧ ¬ 𝐶 ∈ V) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
2221ancoms 459 . . . . . . 7 ((¬ 𝐶 ∈ V ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → {𝐴, 𝐵} ≠ {𝐶, 𝐷})
23 eqneqall 3031 . . . . . . 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 4789 . . . . . . . . 9 (((𝐴𝑉𝐵𝑊𝐴𝐵) ∧ ¬ 𝐷 ∈ V) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
2619, 25sylan 580 . . . . . . . 8 ((((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) ∧ ¬ 𝐷 ∈ V) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
2726ancoms 459 . . . . . . 7 ((¬ 𝐷 ∈ V ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → {𝐴, 𝐵} ≠ {𝐷, 𝐶})
28 prcom 4666 . . . . . . . . 9 {𝐶, 𝐷} = {𝐷, 𝐶}
2928eqeq2i 2837 . . . . . . . 8 ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ {𝐴, 𝐵} = {𝐷, 𝐶})
30 eqneqall 3031 . . . . . . . 8 ({𝐴, 𝐵} = {𝐷, 𝐶} → ({𝐴, 𝐵} ≠ {𝐷, 𝐶} → (𝐴 = 𝐶𝐵 = 𝐷)))
3129, 30sylbi 218 . . . . . . 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 952 . . . . 5 (((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → ({𝐴, 𝐵} = {𝐶, 𝐷} → (𝐴 = 𝐶𝐵 = 𝐷)))
34 preq12 4669 . . . . 5 ((𝐴 = 𝐶𝐵 = 𝐷) → {𝐴, 𝐵} = {𝐶, 𝐷})
3533, 34impbid1 226 . . . 4 (((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) ∧ ((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷))) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
3635ex 413 . . 3 ((¬ 𝐶 ∈ V ∨ ¬ 𝐷 ∈ V) → (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷))))
3715, 36sylbi 218 . 2 (¬ (𝐶 ∈ V ∧ 𝐷 ∈ V) → (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷))))
3814, 37pm2.61i 183 1 (((𝐴𝑉𝐵𝑊) ∧ (𝐴𝐵𝐴𝐷)) → ({𝐴, 𝐵} = {𝐶, 𝐷} ↔ (𝐴 = 𝐶𝐵 = 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 843  w3a 1081   = wceq 1530  wcel 2106  wne 3020  Vcvv 3499  {cpr 4565
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-v 3501  df-dif 3942  df-un 3944  df-nul 4295  df-sn 4564  df-pr 4566
This theorem is referenced by:  linds2eq  30856
  Copyright terms: Public domain W3C validator