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Theorem tpprceq3 4729
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 977 . 2 (¬ (𝐶 ∈ V ∧ 𝐶𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐵))
2 prprc2 4694 . . . . 5 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵})
32uneq1d 4136 . . . 4 𝐶 ∈ V → ({𝐵, 𝐶} ∪ {𝐴}) = ({𝐵} ∪ {𝐴}))
4 tprot 4677 . . . . 5 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
5 df-tp 4564 . . . . 5 {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
64, 5eqtri 2842 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐵, 𝐶} ∪ {𝐴})
7 prcom 4660 . . . . 5 {𝐴, 𝐵} = {𝐵, 𝐴}
8 df-pr 4562 . . . . 5 {𝐵, 𝐴} = ({𝐵} ∪ {𝐴})
97, 8eqtri 2842 . . . 4 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
103, 6, 93eqtr4g 2879 . . 3 𝐶 ∈ V → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
11 nne 3018 . . . 4 𝐶𝐵𝐶 = 𝐵)
12 tppreq3 4687 . . . . 5 (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1312eqcoms 2827 . . . 4 (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1411, 13sylbi 219 . . 3 𝐶𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1510, 14jaoi 853 . 2 ((¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
161, 15sylbi 219 1 (¬ (𝐶 ∈ V ∧ 𝐶𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 398  wo 843   = wceq 1530  wcel 2107  wne 3014  Vcvv 3493  cun 3932  {csn 4559  {cpr 4561  {ctp 4563
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 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2791
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1082  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-v 3495  df-dif 3937  df-un 3939  df-nul 4290  df-sn 4560  df-pr 4562  df-tp 4564
This theorem is referenced by:  tppreqb  4730  1to3vfriswmgr  28051
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