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Mirrors > Home > ILE Home > Th. List > ax-pr | Structured version 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 3851). (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 1374 | . . . . 5 wff w = x |
4 | vy | . . . . . 6 setvar y | |
5 | 1, 4 | weq 1374 | . . . . 5 wff w = y |
6 | 3, 5 | wo 616 | . . . 4 wff (w = x ∨ w = y) |
7 | vz | . . . . 5 setvar z | |
8 | 1, 7 | wel 1376 | . . . 4 wff w ∈ z |
9 | 6, 8 | wi 4 | . . 3 wff ((w = x ∨ w = y) → w ∈ z) |
10 | 9, 1 | wal 1226 | . 2 wff ∀w((w = x ∨ w = y) → w ∈ z) |
11 | 10, 7 | wex 1363 | 1 wff ∃z∀w((w = x ∨ w = y) → w ∈ z) |
Colors of variables: wff set class |
This axiom is referenced by: zfpair2 3918 bj-zfpair2 6477 |
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