Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > zfinf | Structured version Visualization version GIF version |
Description: Axiom of Infinity expressed with the fewest number of different variables. (New usage is discouraged.) (Contributed by NM, 14-Aug-2003.) |
Ref | Expression |
---|---|
zfinf | ⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ax-inf 8418 | . 2 ⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) | |
2 | elequ1 1984 | . . . . . 6 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑥 ↔ 𝑦 ∈ 𝑥)) | |
3 | elequ1 1984 | . . . . . . . 8 ⊢ (𝑤 = 𝑦 → (𝑤 ∈ 𝑧 ↔ 𝑦 ∈ 𝑧)) | |
4 | 3 | anbi1d 737 | . . . . . . 7 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥) ↔ (𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
5 | 4 | exbidv 1837 | . . . . . 6 ⊢ (𝑤 = 𝑦 → (∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥) ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
6 | 2, 5 | imbi12d 333 | . . . . 5 ⊢ (𝑤 = 𝑦 → ((𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) ↔ (𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) |
7 | 6 | cbvalv 2261 | . . . 4 ⊢ (∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) ↔ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
8 | 7 | anbi2i 726 | . . 3 ⊢ ((𝑦 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) ↔ (𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) |
9 | 8 | exbii 1764 | . 2 ⊢ (∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑤(𝑤 ∈ 𝑥 → ∃𝑧(𝑤 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) ↔ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) |
10 | 1, 9 | mpbi 219 | 1 ⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∃wex 1695 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-inf 8418 |
This theorem depends on definitions: df-bi 196 df-an 385 df-ex 1696 df-nf 1701 |
This theorem is referenced by: axinf2 8420 axinfndlem1 9306 |
Copyright terms: Public domain | W3C validator |