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Mirrors > Home > ILE Home > Th. List > zfinf2 | GIF version |
Description: A standard version of the Axiom of Infinity, using definitions to abbreviate. Axiom Inf of [BellMachover] p. 472. (Contributed by NM, 30-Aug-1993.) |
Ref | Expression |
---|---|
zfinf2 | ⊢ ∃x(∅ ∈ x ∧ ∀y ∈ x suc y ∈ x) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-iinf 4254 | . 2 ⊢ ∃x(∅ ∈ x ∧ ∀y(y ∈ x → suc y ∈ x)) | |
2 | df-ral 2305 | . . . 4 ⊢ (∀y ∈ x suc y ∈ x ↔ ∀y(y ∈ x → suc y ∈ x)) | |
3 | 2 | anbi2i 430 | . . 3 ⊢ ((∅ ∈ x ∧ ∀y ∈ x suc y ∈ x) ↔ (∅ ∈ x ∧ ∀y(y ∈ x → suc y ∈ x))) |
4 | 3 | exbii 1493 | . 2 ⊢ (∃x(∅ ∈ x ∧ ∀y ∈ x suc y ∈ x) ↔ ∃x(∅ ∈ x ∧ ∀y(y ∈ x → suc y ∈ x))) |
5 | 1, 4 | mpbir 134 | 1 ⊢ ∃x(∅ ∈ x ∧ ∀y ∈ x suc y ∈ x) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∀wal 1240 ∃wex 1378 ∈ wcel 1390 ∀wral 2300 ∅c0 3218 suc csuc 4068 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-4 1397 ax-ial 1424 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-ral 2305 |
This theorem is referenced by: omex 4259 |
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