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Mirrors > Home > ILE Home > Th. List > pssss | GIF version |
Description: A proper subclass is a subclass. Theorem 10 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) |
Ref | Expression |
---|---|
pssss | ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pss 2933 | . 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)) | |
2 | 1 | simplbi 259 | 1 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ≠ wne 2204 ⊆ wss 2917 ⊊ wpss 2918 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 |
This theorem depends on definitions: df-bi 110 df-pss 2933 |
This theorem is referenced by: pssssd 3041 sspssr 3043 pssn2lp 3045 sspsstrir 3046 psstr 3049 sspsstr 3050 psssstr 3051 |
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