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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 9630 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 9613 | . 2 ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
3 | dfss2 3960 | . . . . 5 ⊢ (𝑥 ⊆ ∪ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝑥)) | |
4 | eluni 4902 | . . . . . . 7 ⊢ (𝑦 ∈ ∪ 𝑥 ↔ ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)) | |
5 | 4 | imbi2i 336 | . . . . . 6 ⊢ ((𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝑥) ↔ (𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
6 | 5 | albii 1813 | . . . . 5 ⊢ (∀𝑦(𝑦 ∈ 𝑥 → 𝑦 ∈ ∪ 𝑥) ↔ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
7 | 3, 6 | bitri 275 | . . . 4 ⊢ (𝑥 ⊆ ∪ 𝑥 ↔ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥))) |
8 | 7 | anbi2i 622 | . . 3 ⊢ ((𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ (𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) |
9 | 8 | exbii 1842 | . 2 ⊢ (∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) ↔ ∃𝑥(𝑥 ≠ ∅ ∧ ∀𝑦(𝑦 ∈ 𝑥 → ∃𝑧(𝑦 ∈ 𝑧 ∧ 𝑧 ∈ 𝑥)))) |
10 | 2, 9 | mpbir 230 | 1 ⊢ ∃𝑥(𝑥 ≠ ∅ ∧ 𝑥 ⊆ ∪ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∀wal 1531 ∃wex 1773 ∈ wcel 2098 ≠ wne 2932 ⊆ wss 3940 ∅c0 4314 ∪ cuni 4899 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-v 3468 df-dif 3943 df-in 3947 df-ss 3957 df-nul 4315 df-uni 4900 |
This theorem is referenced by: axinf2 9631 |
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