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Mirrors > Home > ILE Home > Th. List > dfpss2 | GIF version |
Description: Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) |
Ref | Expression |
---|---|
dfpss2 | ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pss 2933 | . 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵)) | |
2 | df-ne 2206 | . . 3 ⊢ (𝐴 ≠ 𝐵 ↔ ¬ 𝐴 = 𝐵) | |
3 | 2 | anbi2i 430 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐴 ≠ 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) |
4 | 1, 3 | bitri 173 | 1 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ↔ wb 98 = wceq 1243 ≠ wne 2204 ⊆ wss 2917 ⊊ wpss 2918 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 |
This theorem depends on definitions: df-bi 110 df-ne 2206 df-pss 2933 |
This theorem is referenced by: dfpss3 3030 psstr 3049 sspsstr 3050 psssstr 3051 pssv 3267 disj4im 3276 ssnelpss 3289 onpsssuc 4295 f1imapss 5415 |
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