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Mirrors > Home > ILE Home > Th. List > ax-iinf | GIF version |
Description: Axiom of Infinity. Axiom 5 of [Crosilla] p. "Axioms of CZF and IZF". (Contributed by Jim Kingdon, 16-Nov-2018.) |
Ref | Expression |
---|---|
ax-iinf | ⊢ ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | c0 3422 | . . . 4 class ∅ | |
2 | vx | . . . . 5 setvar 𝑥 | |
3 | 2 | cv 1352 | . . . 4 class 𝑥 |
4 | 1, 3 | wcel 2148 | . . 3 wff ∅ ∈ 𝑥 |
5 | vy | . . . . . 6 setvar 𝑦 | |
6 | 5, 2 | wel 2149 | . . . . 5 wff 𝑦 ∈ 𝑥 |
7 | 5 | cv 1352 | . . . . . . 7 class 𝑦 |
8 | 7 | csuc 4365 | . . . . . 6 class suc 𝑦 |
9 | 8, 3 | wcel 2148 | . . . . 5 wff suc 𝑦 ∈ 𝑥 |
10 | 6, 9 | wi 4 | . . . 4 wff (𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥) |
11 | 10, 5 | wal 1351 | . . 3 wff ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥) |
12 | 4, 11 | wa 104 | . 2 wff (∅ ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)) |
13 | 12, 2 | wex 1492 | 1 wff ∃𝑥(∅ ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → suc 𝑦 ∈ 𝑥)) |
Colors of variables: wff set class |
This axiom is referenced by: zfinf2 4588 |
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