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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 483 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴) | |
2 | 1 | abssi 3967 | 1 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∈ wcel 2081 {cab 2775 ⊆ wss 3859 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-in 3866 df-ss 3874 |
This theorem is referenced by: ssrab2 3977 zfausab 5124 exss 5247 dmopabss 5673 fabexg 7495 isf32lem9 9629 psubspset 36411 psubclsetN 36603 |
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