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Theorem tppreqb 4305
Description: An unordered triple is an unordered pair if and only if one of its elements is a proper class or is identical with one of the another elements. (Contributed by Alexander van der Vekens, 15-Jan-2018.)
Assertion
Ref Expression
tppreqb (¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵) ↔ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Proof of Theorem tppreqb
StepHypRef Expression
1 3ianor 1053 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
2 df-3or 1037 . . . 4 ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵) ↔ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ ¬ 𝐶𝐵))
31, 2bitri 264 . . 3 (¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵) ↔ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ ¬ 𝐶𝐵))
4 orass 546 . . . . 5 ((((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ ¬ 𝐶𝐵) ∨ ¬ 𝐶 ∈ V) ↔ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ (¬ 𝐶𝐵 ∨ ¬ 𝐶 ∈ V)))
5 ianor 509 . . . . . . . 8 (¬ (𝐶 ∈ V ∧ 𝐶𝐴) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴))
6 tpprceq3 4304 . . . . . . . 8 (¬ (𝐶 ∈ V ∧ 𝐶𝐴) → {𝐵, 𝐴, 𝐶} = {𝐵, 𝐴})
75, 6sylbir 225 . . . . . . 7 ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) → {𝐵, 𝐴, 𝐶} = {𝐵, 𝐴})
8 tpcoma 4255 . . . . . . 7 {𝐵, 𝐴, 𝐶} = {𝐴, 𝐵, 𝐶}
9 prcom 4237 . . . . . . 7 {𝐵, 𝐴} = {𝐴, 𝐵}
107, 8, 93eqtr3g 2678 . . . . . 6 ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
11 orcom 402 . . . . . . . 8 ((¬ 𝐶𝐵 ∨ ¬ 𝐶 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐵))
12 ianor 509 . . . . . . . 8 (¬ (𝐶 ∈ V ∧ 𝐶𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐵))
1311, 12bitr4i 267 . . . . . . 7 ((¬ 𝐶𝐵 ∨ ¬ 𝐶 ∈ V) ↔ ¬ (𝐶 ∈ V ∧ 𝐶𝐵))
14 tpprceq3 4304 . . . . . . 7 (¬ (𝐶 ∈ V ∧ 𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1513, 14sylbi 207 . . . . . 6 ((¬ 𝐶𝐵 ∨ ¬ 𝐶 ∈ V) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1610, 15jaoi 394 . . . . 5 (((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ (¬ 𝐶𝐵 ∨ ¬ 𝐶 ∈ V)) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
174, 16sylbi 207 . . . 4 ((((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ ¬ 𝐶𝐵) ∨ ¬ 𝐶 ∈ V) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1817orcs 409 . . 3 (((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ ¬ 𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
193, 18sylbi 207 . 2 (¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
20 df-tp 4153 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
2120eqeq1i 2626 . . 3 ({𝐴, 𝐵, 𝐶} = {𝐴, 𝐵} ↔ ({𝐴, 𝐵} ∪ {𝐶}) = {𝐴, 𝐵})
22 ssequn2 3764 . . . 4 ({𝐶} ⊆ {𝐴, 𝐵} ↔ ({𝐴, 𝐵} ∪ {𝐶}) = {𝐴, 𝐵})
23 snssg 4296 . . . . . . 7 (𝐶 ∈ V → (𝐶 ∈ {𝐴, 𝐵} ↔ {𝐶} ⊆ {𝐴, 𝐵}))
24 elpri 4168 . . . . . . . 8 (𝐶 ∈ {𝐴, 𝐵} → (𝐶 = 𝐴𝐶 = 𝐵))
25 nne 2794 . . . . . . . . . 10 𝐶𝐴𝐶 = 𝐴)
26 3mix2 1229 . . . . . . . . . 10 𝐶𝐴 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
2725, 26sylbir 225 . . . . . . . . 9 (𝐶 = 𝐴 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
28 nne 2794 . . . . . . . . . 10 𝐶𝐵𝐶 = 𝐵)
29 3mix3 1230 . . . . . . . . . 10 𝐶𝐵 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
3028, 29sylbir 225 . . . . . . . . 9 (𝐶 = 𝐵 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
3127, 30jaoi 394 . . . . . . . 8 ((𝐶 = 𝐴𝐶 = 𝐵) → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
3224, 31syl 17 . . . . . . 7 (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
3323, 32syl6bir 244 . . . . . 6 (𝐶 ∈ V → ({𝐶} ⊆ {𝐴, 𝐵} → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵)))
34 3mix1 1228 . . . . . . 7 𝐶 ∈ V → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
3534a1d 25 . . . . . 6 𝐶 ∈ V → ({𝐶} ⊆ {𝐴, 𝐵} → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵)))
3633, 35pm2.61i 176 . . . . 5 ({𝐶} ⊆ {𝐴, 𝐵} → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
3736, 1sylibr 224 . . . 4 ({𝐶} ⊆ {𝐴, 𝐵} → ¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵))
3822, 37sylbir 225 . . 3 (({𝐴, 𝐵} ∪ {𝐶}) = {𝐴, 𝐵} → ¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵))
3921, 38sylbi 207 . 2 ({𝐴, 𝐵, 𝐶} = {𝐴, 𝐵} → ¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵))
4019, 39impbii 199 1 (¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵) ↔ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 383  wa 384  w3o 1035  w3a 1036   = wceq 1480  wcel 1987  wne 2790  Vcvv 3186  cun 3553  wss 3555  {csn 4148  {cpr 4150  {ctp 4152
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-sn 4149  df-pr 4151  df-tp 4153
This theorem is referenced by: (None)
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