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Theorem tpprceq3 4471
Description: An unordered triple is an unordered pair if one of its elements is a proper class or is identical with another element. (Contributed by Alexander van der Vekens, 6-Oct-2017.)
Assertion
Ref Expression
tpprceq3 (¬ (𝐶 ∈ V ∧ 𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Proof of Theorem tpprceq3
StepHypRef Expression
1 ianor 966 . 2 (¬ (𝐶 ∈ V ∧ 𝐶𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐵))
2 prprc2 4438 . . . . 5 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵})
32uneq1d 3917 . . . 4 𝐶 ∈ V → ({𝐵, 𝐶} ∪ {𝐴}) = ({𝐵} ∪ {𝐴}))
4 tprot 4421 . . . . 5 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
5 df-tp 4322 . . . . 5 {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
64, 5eqtri 2793 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐵, 𝐶} ∪ {𝐴})
7 prcom 4404 . . . . 5 {𝐴, 𝐵} = {𝐵, 𝐴}
8 df-pr 4320 . . . . 5 {𝐵, 𝐴} = ({𝐵} ∪ {𝐴})
97, 8eqtri 2793 . . . 4 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
103, 6, 93eqtr4g 2830 . . 3 𝐶 ∈ V → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
11 nne 2947 . . . 4 𝐶𝐵𝐶 = 𝐵)
12 tppreq3 4431 . . . . 5 (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1312eqcoms 2779 . . . 4 (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1411, 13sylbi 207 . . 3 𝐶𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1510, 14jaoi 846 . 2 ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
161, 15sylbi 207 1 (¬ (𝐶 ∈ V ∧ 𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 382  wo 836   = wceq 1631  wcel 2145  wne 2943  Vcvv 3351  cun 3721  {csn 4317  {cpr 4319  {ctp 4321
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-3or 1072  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 4318  df-pr 4320  df-tp 4322
This theorem is referenced by:  tppreqb  4472  1to3vfriswmgr  27462
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