Theorem tpprceq3 4764

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 995	. 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵))
2		prprc2 4725	. . . . 5 ⊢ (¬ 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵})
3	2	uneq1d 4120	. . . 4 ⊢ (¬ 𝐶 ∈ V → ({𝐵, 𝐶} ∪ {𝐴}) = ({𝐵} ∪ {𝐴}))
4		tprot 4708	. . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
5		df-tp 4587	. . . . 5 ⊢ {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
6	4, 5	eqtri 2785	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐵, 𝐶} ∪ {𝐴})
7		prcom 4691	. . . . 5 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
8		df-pr 4585	. . . . 5 ⊢ {𝐵, 𝐴} = ({𝐵} ∪ {𝐴})
9	7, 8	eqtri 2785	. . . 4 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
10	3, 6, 9	3eqtr4g 2822	. . 3 ⊢ (¬ 𝐶 ∈ V → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
11		nne 2961	. . . 4 ⊢ (¬ 𝐶 ≠ 𝐵 ↔ 𝐶 = 𝐵)
12		tppreq3 4718	. . . . 5 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
13	12	eqcoms 2770	. . . 4 ⊢ (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
14	11, 13	sylbi 219	. . 3 ⊢ (¬ 𝐶 ≠ 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
15	10, 14	jaoi 868	. 2 ⊢ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
16	1, 15	sylbi 219	1 ⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858   = wceq 1560   ∈ wcel 2142   ≠ wne 2957  Vcvv 3454   ∪ cun 3902  {csn 4582  {cpr 4584  {ctp 4586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1099  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-v 3456  df-dif 3907  df-un 3909  df-nul 4286  df-sn 4583  df-pr 4585  df-tp 4587

This theorem is referenced by:  tppreqb  4765  1to3vfriswmgr  30479  tpssad  32735