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Theorem eltp 3771

Description: A member of an unordered triple of classes is one of them. Special case of Exercise 1 of [TakeutiZaring] p. 17. (Contributed by NM, 8-Apr-1994.) (Revised by Mario Carneiro, 11-Feb-2015.)

**Hypothesis**
Ref	Expression
eltp.1	⊢ A ∈ V

**Assertion**
Ref	Expression
eltp	⊢ (A ∈ {B, C, D} ↔ (A = B ∨ A = C ∨ A = D))

**Proof of Theorem eltp**
Step	Hyp	Ref	Expression
1		eltp.1	. 2 ⊢ A ∈ V
2		eltpg 3769	. 2 ⊢ (A ∈ V → (A ∈ {B, C, D} ↔ (A = B ∨ A = C ∨ A = D)))
3	1, 2	ax-mp 8	1 ⊢ (A ∈ {B, C, D} ↔ (A = B ∨ A = C ∨ A = D))

Colors of variables: wff setvar class

Syntax hints: ↔ wb 176 ∨ w3o 933 = wceq 1642 ∈ wcel 1710 Vcvv 2859 {ctp 3739

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334

This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-3or 935 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-sn 3741 df-pr 3742 df-tp 3743

This theorem is referenced by: dftp2 3772 tpid1 3829 tpid2 3830 tpid3 3832

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