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Mirrors > Home > ILE Home > Th. List > ax-pr | GIF version |
Description: The Axiom of Pairing of IZF set theory. Axiom 2 of [Crosilla] p. "Axioms of CZF and IZF", except (a) unnecessary quantifiers are removed, and (b) Crosilla has a biconditional rather than an implication (but the two are equivalent by bm1.3ii 4108). (Contributed by NM, 14-Nov-2006.) |
Ref | Expression |
---|---|
ax-pr | ⊢ ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vw | . . . . . 6 setvar 𝑤 | |
2 | vx | . . . . . 6 setvar 𝑥 | |
3 | 1, 2 | weq 1496 | . . . . 5 wff 𝑤 = 𝑥 |
4 | vy | . . . . . 6 setvar 𝑦 | |
5 | 1, 4 | weq 1496 | . . . . 5 wff 𝑤 = 𝑦 |
6 | 3, 5 | wo 703 | . . . 4 wff (𝑤 = 𝑥 ∨ 𝑤 = 𝑦) |
7 | vz | . . . . 5 setvar 𝑧 | |
8 | 1, 7 | wel 2142 | . . . 4 wff 𝑤 ∈ 𝑧 |
9 | 6, 8 | wi 4 | . . 3 wff ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
10 | 9, 1 | wal 1346 | . 2 wff ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
11 | 10, 7 | wex 1485 | 1 wff ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
Colors of variables: wff set class |
This axiom is referenced by: zfpair2 4193 bj-zfpair2 13867 |
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