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Mirrors > Home > MPE Home > Th. List > abanssl | Structured version Visualization version GIF version |
Description: A class abstraction with a conjunction is a subset of the class abstraction with the left conjunct only. (Contributed by AV, 7-Aug-2024.) (Proof shortened by SN, 22-Aug-2024.) |
Ref | Expression |
---|---|
abanssl | ⊢ {𝑓 ∣ (𝜑 ∧ 𝜓)} ⊆ {𝑓 ∣ 𝜑} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 482 | . 2 ⊢ ((𝜑 ∧ 𝜓) → 𝜑) | |
2 | 1 | ss2abi 4058 | 1 ⊢ {𝑓 ∣ (𝜑 ∧ 𝜓)} ⊆ {𝑓 ∣ 𝜑} |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 395 {cab 2703 ⊆ wss 3943 |
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-9 2108 ax-ext 2697 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2704 df-cleq 2718 df-in 3950 df-ss 3960 |
This theorem is referenced by: f1setex 8850 fsetprcnexALT 46325 |
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