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Mirrors > Home > MPE Home > Th. List > opex | Structured version Visualization version GIF version |
Description: An ordered pair of classes is a set. Exercise 7 of [TakeutiZaring] p. 16. (Contributed by NM, 18-Aug-1993.) (Revised by Mario Carneiro, 26-Apr-2015.) |
Ref | Expression |
---|---|
opex | ⊢ ⟨𝐴, 𝐵⟩ ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfopif 4828 | . 2 ⊢ ⟨𝐴, 𝐵⟩ = if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) | |
2 | prex 5390 | . . 3 ⊢ {{𝐴}, {𝐴, 𝐵}} ∈ V | |
3 | 0ex 5265 | . . 3 ⊢ ∅ ∈ V | |
4 | 2, 3 | ifex 4537 | . 2 ⊢ if((𝐴 ∈ V ∧ 𝐵 ∈ V), {{𝐴}, {𝐴, 𝐵}}, ∅) ∈ V |
5 | 1, 4 | eqeltri 2830 | 1 ⊢ ⟨𝐴, 𝐵⟩ ∈ V |
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