Theorem tpprceq3 4804

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 ∧ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Proof of Theorem tpprceq3

Step	Hyp	Ref	Expression
1		ianor 980	. 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵))
2		prprc2 4767	. . . . 5 ⊢ (¬ 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵})
3	2	uneq1d 4159	. . . 4 ⊢ (¬ 𝐶 ∈ V → ({𝐵, 𝐶} ∪ {𝐴}) = ({𝐵} ∪ {𝐴}))
4		tprot 4750	. . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
5		df-tp 4630	. . . . 5 ⊢ {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
6	4, 5	eqtri 2756	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐵, 𝐶} ∪ {𝐴})
7		prcom 4733	. . . . 5 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
8		df-pr 4628	. . . . 5 ⊢ {𝐵, 𝐴} = ({𝐵} ∪ {𝐴})
9	7, 8	eqtri 2756	. . . 4 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
10	3, 6, 9	3eqtr4g 2793	. . 3 ⊢ (¬ 𝐶 ∈ V → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
11		nne 2940	. . . 4 ⊢ (¬ 𝐶 ≠ 𝐵 ↔ 𝐶 = 𝐵)
12		tppreq3 4760	. . . . 5 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
13	12	eqcoms 2736	. . . 4 ⊢ (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
14	11, 13	sylbi 216	. . 3 ⊢ (¬ 𝐶 ≠ 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
15	10, 14	jaoi 856	. 2 ⊢ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
16	1, 15	sylbi 216	1 ⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 846   = wceq 1534   ∈ wcel 2099   ≠ wne 2936  Vcvv 3470   ∪ cun 3943  {csn 4625  {cpr 4627  {ctp 4629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ne 2937  df-v 3472  df-dif 3948  df-un 3950  df-nul 4320  df-sn 4626  df-pr 4628  df-tp 4630

This theorem is referenced by:  tppreqb  4805  1to3vfriswmgr  30084