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Mirrors > Home > ILE Home > Th. List > pssirr | GIF version |
Description: Proper subclass is irreflexive. Theorem 7 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) |
Ref | Expression |
---|---|
pssirr | ⊢ ¬ 𝐴 ⊊ 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pm3.24 627 | . 2 ⊢ ¬ (𝐴 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐴) | |
2 | dfpss3 3030 | . 2 ⊢ (𝐴 ⊊ 𝐴 ↔ (𝐴 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐴)) | |
3 | 1, 2 | mtbir 596 | 1 ⊢ ¬ 𝐴 ⊊ 𝐴 |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ∧ wa 97 ⊆ 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: (None) |
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