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Mirrors > Home > ILE Home > Th. List > dfpss3 | GIF version |
Description: Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
dfpss3 | ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfpss2 3029 | . 2 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵)) | |
2 | eqss 2960 | . . . . 5 ⊢ (𝐴 = 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐴)) | |
3 | 2 | baib 828 | . . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 = 𝐵 ↔ 𝐵 ⊆ 𝐴)) |
4 | 3 | notbid 592 | . . 3 ⊢ (𝐴 ⊆ 𝐵 → (¬ 𝐴 = 𝐵 ↔ ¬ 𝐵 ⊆ 𝐴)) |
5 | 4 | pm5.32i 427 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ¬ 𝐴 = 𝐵) ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) |
6 | 1, 5 | bitri 173 | 1 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ↔ wb 98 = wceq 1243 ⊆ 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 ax-in1 544 ax-in2 545 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-ne 2206 df-in 2924 df-ss 2931 df-pss 2933 |
This theorem is referenced by: pssirr 3044 pssn2lp 3045 ssnpss 3047 nsspssun 3170 npss0 3266 |
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