Theorem tpprceq3 4772

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 997	. 2 ⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) ↔ (¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵))
2		prprc2 4732	. . . . 5 ⊢ (¬ 𝐶 ∈ V → {𝐵, 𝐶} = {𝐵})
3	2	uneq1d 4121	. . . 4 ⊢ (¬ 𝐶 ∈ V → ({𝐵, 𝐶} ∪ {𝐴}) = ({𝐵} ∪ {𝐴}))
4		tprot 4715	. . . . 5 ⊢ {𝐴, 𝐵, 𝐶} = {𝐵, 𝐶, 𝐴}
5		df-tp 4594	. . . . 5 ⊢ {𝐵, 𝐶, 𝐴} = ({𝐵, 𝐶} ∪ {𝐴})
6	4, 5	eqtri 2786	. . . 4 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐵, 𝐶} ∪ {𝐴})
7		prcom 4698	. . . . 5 ⊢ {𝐴, 𝐵} = {𝐵, 𝐴}
8		df-pr 4592	. . . . 5 ⊢ {𝐵, 𝐴} = ({𝐵} ∪ {𝐴})
9	7, 8	eqtri 2786	. . . 4 ⊢ {𝐴, 𝐵} = ({𝐵} ∪ {𝐴})
10	3, 6, 9	3eqtr4g 2823	. . 3 ⊢ (¬ 𝐶 ∈ V → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
11		nne 2962	. . . 4 ⊢ (¬ 𝐶 ≠ 𝐵 ↔ 𝐶 = 𝐵)
12		tppreq3 4725	. . . . 5 ⊢ (𝐵 = 𝐶 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
13	12	eqcoms 2771	. . . 4 ⊢ (𝐶 = 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
14	11, 13	sylbi 220	. . 3 ⊢ (¬ 𝐶 ≠ 𝐵 → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
15	10, 14	jaoi 870	. 2 ⊢ ((¬ 𝐶 ∈ V ∨ ¬ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})
16	1, 15	sylbi 220	1 ⊢ (¬ (𝐶 ∈ V ∧ 𝐶 ≠ 𝐵) → {𝐴, 𝐵, 𝐶} = {𝐴, 𝐵})

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   = wceq 1570   ∈ wcel 2143   ≠ wne 2958  Vcvv 3455   ∪ cun 3903  {csn 4589  {cpr 4591  {ctp 4593

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-v 3457  df-dif 3908  df-un 3910  df-nul 4287  df-sn 4590  df-pr 4592  df-tp 4594

This theorem is referenced by:  tppreqb  4773  1to3vfriswmgr  30631  tpssad  32885