Theorem elpr2 4422

Description: A member of an unordered pair of classes is one or the other of them. Exercise 1 of [TakeutiZaring] p. 15. (Contributed by NM, 14-Oct-2005.) (Proof shortened by JJ, 23-Jul-2021.)

Hypotheses

Ref	Expression
elpr2.1	⊢ 𝐵 ∈ V
elpr2.2	⊢ 𝐶 ∈ V

Assertion

Ref	Expression
elpr2	⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))

Proof of Theorem elpr2

Step	Hyp	Ref	Expression
1		elex 3414	. 2 ⊢ (𝐴 ∈ {𝐵, 𝐶} → 𝐴 ∈ V)
2		elpr2.1	. . . 4 ⊢ 𝐵 ∈ V
3		eleq1 2847	. . . 4 ⊢ (𝐴 = 𝐵 → (𝐴 ∈ V ↔ 𝐵 ∈ V))
4	2, 3	mpbiri 250	. . 3 ⊢ (𝐴 = 𝐵 → 𝐴 ∈ V)
5		elpr2.2	. . . 4 ⊢ 𝐶 ∈ V
6		eleq1 2847	. . . 4 ⊢ (𝐴 = 𝐶 → (𝐴 ∈ V ↔ 𝐶 ∈ V))
7	5, 6	mpbiri 250	. . 3 ⊢ (𝐴 = 𝐶 → 𝐴 ∈ V)
8	4, 7	jaoi 846	. 2 ⊢ ((𝐴 = 𝐵 ∨ 𝐴 = 𝐶) → 𝐴 ∈ V)
9		elprg 4419	. 2 ⊢ (𝐴 ∈ V → (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶)))
10	1, 8, 9	pm5.21nii 370	1 ⊢ (𝐴 ∈ {𝐵, 𝐶} ↔ (𝐴 = 𝐵 ∨ 𝐴 = 𝐶))

Syntax hints:   ↔ wb 198   ∨ wo 836   = wceq 1601   ∈ wcel 2107  Vcvv 3398  {cpr 4400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-ext 2754

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-v 3400  df-un 3797  df-sn 4399  df-pr 4401

This theorem is referenced by:  elopg  5166  elxr  12261  fprodex01  30135  nofv  32399