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Theorem tppreqb 4490
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 1132 . . . 4 (¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
2 df-3or 1108 . . . 4 ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵) ↔ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ ¬ 𝐶𝐵))
31, 2bitri 266 . . 3 (¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵) ↔ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ ¬ 𝐶𝐵))
4 orass 945 . . . . 5 ((((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ ¬ 𝐶𝐵) ∨ ¬ 𝐶 ∈ V) ↔ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ (¬ 𝐶𝐵 ∨ ¬ 𝐶 ∈ V)))
5 ianor 1004 . . . . . . . 8 (¬ (𝐶 ∈ V ∧ 𝐶𝐴) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴))
6 tpprceq3 4489 . . . . . . . 8 (¬ (𝐶 ∈ V ∧ 𝐶𝐴) → {𝐵, 𝐴, 𝐶} = {𝐵, 𝐴})
75, 6sylbir 226 . . . . . . 7 ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) → {𝐵, 𝐴, 𝐶} = {𝐵, 𝐴})
8 tpcoma 4440 . . . . . . 7 {𝐵, 𝐴, 𝐶} = {𝐴, 𝐵, 𝐶}
9 prcom 4422 . . . . . . 7 {𝐵, 𝐴} = {𝐴, 𝐵}
107, 8, 93eqtr3g 2822 . . . . . 6 ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
11 orcom 896 . . . . . . . 8 ((¬ 𝐶𝐵 ∨ ¬ 𝐶 ∈ V) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐵))
12 ianor 1004 . . . . . . . 8 (¬ (𝐶 ∈ V ∧ 𝐶𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐵))
1311, 12bitr4i 269 . . . . . . 7 ((¬ 𝐶𝐵 ∨ ¬ 𝐶 ∈ V) ↔ ¬ (𝐶 ∈ V ∧ 𝐶𝐵))
14 tpprceq3 4489 . . . . . . 7 (¬ (𝐶 ∈ V ∧ 𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1513, 14sylbi 208 . . . . . 6 ((¬ 𝐶𝐵 ∨ ¬ 𝐶 ∈ V) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1610, 15jaoi 883 . . . . 5 (((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ (¬ 𝐶𝐵 ∨ ¬ 𝐶 ∈ V)) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
174, 16sylbi 208 . . . 4 ((((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ ¬ 𝐶𝐵) ∨ ¬ 𝐶 ∈ V) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1817orcs 901 . . 3 (((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴) ∨ ¬ 𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
193, 18sylbi 208 . 2 (¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
20 df-tp 4339 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
2120eqeq1i 2770 . . 3 ({𝐴, 𝐵, 𝐶} = {𝐴, 𝐵} ↔ ({𝐴, 𝐵} ∪ {𝐶}) = {𝐴, 𝐵})
22 ssequn2 3948 . . . 4 ({𝐶} ⊆ {𝐴, 𝐵} ↔ ({𝐴, 𝐵} ∪ {𝐶}) = {𝐴, 𝐵})
23 snssg 4469 . . . . . . 7 (𝐶 ∈ V → (𝐶 ∈ {𝐴, 𝐵} ↔ {𝐶} ⊆ {𝐴, 𝐵}))
24 elpri 4356 . . . . . . . 8 (𝐶 ∈ {𝐴, 𝐵} → (𝐶 = 𝐴𝐶 = 𝐵))
25 nne 2941 . . . . . . . . . 10 𝐶𝐴𝐶 = 𝐴)
26 3mix2 1430 . . . . . . . . . 10 𝐶𝐴 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
2725, 26sylbir 226 . . . . . . . . 9 (𝐶 = 𝐴 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
28 nne 2941 . . . . . . . . . 10 𝐶𝐵𝐶 = 𝐵)
29 3mix3 1431 . . . . . . . . . 10 𝐶𝐵 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
3028, 29sylbir 226 . . . . . . . . 9 (𝐶 = 𝐵 → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
3127, 30jaoi 883 . . . . . . . 8 ((𝐶 = 𝐴𝐶 = 𝐵) → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
3224, 31syl 17 . . . . . . 7 (𝐶 ∈ {𝐴, 𝐵} → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵))
3323, 32syl6bir 245 . . . . . 6 (𝐶 ∈ V → ({𝐶} ⊆ {𝐴, 𝐵} → (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐴 ∨ ¬ 𝐶𝐵)))
34 3mix1 1429 . . . . . . 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 225 . . . 4 ({𝐶} ⊆ {𝐴, 𝐵} → ¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵))
3822, 37sylbir 226 . . 3 (({𝐴, 𝐵} ∪ {𝐶}) = {𝐴, 𝐵} → ¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵))
3921, 38sylbi 208 . 2 ({𝐴, 𝐵, 𝐶} = {𝐴, 𝐵} → ¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵))
4019, 39impbii 200 1 (¬ (𝐶 ∈ V ∧ 𝐶𝐴𝐶𝐵) ↔ {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 197  wa 384  wo 873  w3o 1106  w3a 1107   = wceq 1652  wcel 2155  wne 2937  Vcvv 3350  cun 3730  wss 3732  {csn 4334  {cpr 4336  {ctp 4338
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-v 3352  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-sn 4335  df-pr 4337  df-tp 4339
This theorem is referenced by: (None)
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