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Mirrors > Home > MPE Home > Th. List > ssab2 | Structured version Visualization version GIF version |
Description: Subclass relation for the restriction of a class abstraction. (Contributed by NM, 31-Mar-1995.) |
Ref | Expression |
---|---|
ssab2 | ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 485 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴) | |
2 | 1 | abssi 4046 | 1 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 398 ∈ wcel 2114 {cab 2799 ⊆ wss 3936 |
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 |
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-in 3943 df-ss 3952 |
This theorem is referenced by: ssrab2 4056 zfausab 5233 exss 5355 dmopabss 5787 fabexg 7639 isf32lem9 9783 psubspset 36895 psubclsetN 37087 |
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