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Mirrors > Home > NFE 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 3261 | . 2 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ A ≠ B)) | |
2 | df-ne 2518 | . . 3 ⊢ (A ≠ B ↔ ¬ A = B) | |
3 | 2 | anbi2i 675 | . 2 ⊢ ((A ⊆ B ∧ A ≠ B) ↔ (A ⊆ B ∧ ¬ A = B)) |
4 | 1, 3 | bitri 240 | 1 ⊢ (A ⊊ B ↔ (A ⊆ B ∧ ¬ A = B)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 176 ∧ wa 358 = wceq 1642 ≠ wne 2516 ⊆ wss 3257 ⊊ wpss 3258 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 |
This theorem depends on definitions: df-bi 177 df-an 360 df-ne 2518 df-pss 3261 |
This theorem is referenced by: dfpss3 3355 sspss 3368 psstr 3373 npss 3379 pssv 3590 disj4 3599 ssnelpss 3613 sfinltfin 4535 |
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