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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 482 | . 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝜑) → 𝑥 ∈ 𝐴) | |
2 | 1 | abssi 4060 | 1 ⊢ {𝑥 ∣ (𝑥 ∈ 𝐴 ∧ 𝜑)} ⊆ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 ∈ wcel 2098 {cab 2701 ⊆ wss 3941 |
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-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-v 3468 df-in 3948 df-ss 3958 |
This theorem is referenced by: ssrab2OLD 4071 zfausab 5321 exss 5454 dmopabss 5909 fabexg 7919 isf32lem9 10353 psubspset 39109 psubclsetN 39301 |
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