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Mirrors > Home > MPE Home > Th. List > ax-pr | Structured version Visualization version GIF version |
Description: The Axiom of Pairing of ZF set theory. It was derived as Theorem axpr 5355 above and is therefore redundant, but we state it as a separate axiom here so that its uses can be identified more easily. (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 1970 | . . . . 5 wff 𝑤 = 𝑥 |
4 | vy | . . . . . 6 setvar 𝑦 | |
5 | 1, 4 | weq 1970 | . . . . 5 wff 𝑤 = 𝑦 |
6 | 3, 5 | wo 844 | . . . 4 wff (𝑤 = 𝑥 ∨ 𝑤 = 𝑦) |
7 | vz | . . . . 5 setvar 𝑧 | |
8 | 1, 7 | wel 2111 | . . . 4 wff 𝑤 ∈ 𝑧 |
9 | 6, 8 | wi 4 | . . 3 wff ((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
10 | 9, 1 | wal 1540 | . 2 wff ∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
11 | 10, 7 | wex 1786 | 1 wff ∃𝑧∀𝑤((𝑤 = 𝑥 ∨ 𝑤 = 𝑦) → 𝑤 ∈ 𝑧) |
Colors of variables: wff setvar class |
This axiom is referenced by: zfpair2 5357 el 5361 |
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