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Theorem tpprceq3 4488
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 1004 . 2 (¬ (𝐶 ∈ V ∧ 𝐶𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐵))
2 prprc2 4455 . . . . 5 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵})
32uneq1d 3927 . . . 4 𝐶 ∈ V → ({𝐵, 𝐶} ∪ {𝐴}) = ({𝐵} ∪ {𝐴}))
4 tprot 4438 . . . . 5 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
5 df-tp 4338 . . . . 5 {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
64, 5eqtri 2786 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐵, 𝐶} ∪ {𝐴})
7 prcom 4421 . . . . 5 {𝐴, 𝐵} = {𝐵, 𝐴}
8 df-pr 4336 . . . . 5 {𝐵, 𝐴} = ({𝐵} ∪ {𝐴})
97, 8eqtri 2786 . . . 4 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
103, 6, 93eqtr4g 2823 . . 3 𝐶 ∈ V → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
11 nne 2940 . . . 4 𝐶𝐵𝐶 = 𝐵)
12 tppreq3 4448 . . . . 5 (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1312eqcoms 2772 . . . 4 (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1411, 13sylbi 208 . . 3 𝐶𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1510, 14jaoi 883 . 2 ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
161, 15sylbi 208 1 (¬ (𝐶 ∈ V ∧ 𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384  wo 873   = wceq 1652  wcel 2155  wne 2936  Vcvv 3349  cun 3729  {csn 4333  {cpr 4335  {ctp 4337
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 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2349  ax-ext 2742
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-ne 2937  df-v 3351  df-dif 3734  df-un 3736  df-nul 4079  df-sn 4334  df-pr 4336  df-tp 4338
This theorem is referenced by:  tppreqb  4489  1to3vfriswmgr  27517
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