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Theorem tpprceq3 4306
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 509 . 2 (¬ (𝐶 ∈ V ∧ 𝐶𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶𝐵))
2 prprc2 4273 . . . . 5 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵})
32uneq1d 3746 . . . 4 𝐶 ∈ V → ({𝐵, 𝐶} ∪ {𝐴}) = ({𝐵} ∪ {𝐴}))
4 tprot 4256 . . . . 5 {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
5 df-tp 4155 . . . . 5 {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
64, 5eqtri 2643 . . . 4 {𝐴, 𝐵, 𝐶} = ({𝐵, 𝐶} ∪ {𝐴})
7 prcom 4239 . . . . 5 {𝐴, 𝐵} = {𝐵, 𝐴}
8 df-pr 4153 . . . . 5 {𝐵, 𝐴} = ({𝐵} ∪ {𝐴})
97, 8eqtri 2643 . . . 4 {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
103, 6, 93eqtr4g 2680 . . 3 𝐶 ∈ V → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
11 nne 2794 . . . 4 𝐶𝐵𝐶 = 𝐵)
12 tppreq3 4266 . . . . 5 (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1312eqcoms 2629 . . . 4 (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1411, 13sylbi 207 . . 3 𝐶𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
1510, 14jaoi 394 . 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  wo 383  wa 384   = wceq 1480  wcel 1987  wne 2790  Vcvv 3186  cun 3554  {csn 4150  {cpr 4152  {ctp 4154
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-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 3559  df-un 3561  df-nul 3894  df-sn 4151  df-pr 4153  df-tp 4155
This theorem is referenced by:  tppreqb  4307  1to3vfriswmgr  27015
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