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Mirrors > Home > MPE Home > Th. List > inf2 | Structured version Visualization version GIF version |
Description: Variation of Axiom of Infinity. There exists a nonempty set that is a subset of its union (using zfinf 8419 as a hypothesis). Abbreviated version of the Axiom of Infinity in [FreydScedrov] p. 283. (Contributed by NM, 28-Oct-1996.) |
Ref | Expression |
---|---|
inf1.1 | ⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
Ref | Expression |
---|---|
inf2 | ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inf1.1 | . . 3 ⊢ ∃𝑥(𝑦 ∈ 𝑥 ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) | |
2 | 1 | inf1 8402 | . 2 ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
3 | dfss2 3557 | . . . . 5 ⊢ (𝑥 ⊆ ∪ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝑥)) | |
4 | eluni 4375 | . . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝑥 ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) | |
5 | 4 | imbi2i 325 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝑥) ↔ (𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
6 | 5 | albii 1737 | . . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝑥) ↔ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
7 | 3, 6 | bitri 263 | . . . 4 ⊢ (𝑥 ⊆ ∪ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
8 | 7 | anbi2i 726 | . . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ (𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) |
9 | 8 | exbii 1764 | . 2 ⊢ (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) |
10 | 2, 9 | mpbir 220 | 1 ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∀wal 1473 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ⊆ wss 3540 ∅c0 3874 ∪ cuni 4372 |
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-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-v 3175 df-dif 3543 df-in 3547 df-ss 3554 df-nul 3875 df-uni 4373 |
This theorem is referenced by: axinf2 8420 |
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