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Mirrors > Home > MPE Home > Th. List > al0ssb | Structured version Visualization version GIF version |
Description: The empty set is the unique class which is a subclass of any set. (Contributed by AV, 24-Aug-2022.) |
Ref | Expression |
---|---|
al0ssb | ⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5211 | . . 3 ⊢ ∅ ∈ V | |
2 | sseq2 3993 | . . . 4 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ ∅)) | |
3 | ss0b 4351 | . . . 4 ⊢ (𝑋 ⊆ ∅ ↔ 𝑋 = ∅) | |
4 | 2, 3 | syl6bb 289 | . . 3 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅)) |
5 | 1, 4 | spcv 3606 | . 2 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 → 𝑋 = ∅) |
6 | 0ss 4350 | . . . 4 ⊢ ∅ ⊆ 𝑦 | |
7 | 6 | ax-gen 1796 | . . 3 ⊢ ∀𝑦∅ ⊆ 𝑦 |
8 | sseq1 3992 | . . . 4 ⊢ (𝑋 = ∅ → (𝑋 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦)) | |
9 | 8 | albidv 1921 | . . 3 ⊢ (𝑋 = ∅ → (∀𝑦 𝑋 ⊆ 𝑦 ↔ ∀𝑦∅ ⊆ 𝑦)) |
10 | 7, 9 | mpbiri 260 | . 2 ⊢ (𝑋 = ∅ → ∀𝑦 𝑋 ⊆ 𝑦) |
11 | 5, 10 | impbii 211 | 1 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 ∀wal 1535 = wceq 1537 ⊆ wss 3936 ∅c0 4291 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-nul 5210 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-dif 3939 df-in 3943 df-ss 3952 df-nul 4292 |
This theorem is referenced by: iota0def 43322 aiota0def 43343 |
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