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Mirrors > Home > ILE Home > Th. List > pssn2lp | GIF version |
Description: Proper subclass has no 2-cycle loops. Compare Theorem 8 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) |
Ref | Expression |
---|---|
pssn2lp | ⊢ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfpss3 3030 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) | |
2 | 1 | simprbi 260 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊆ 𝐴) |
3 | pssss 3039 | . . 3 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴) | |
4 | 2, 3 | nsyl 558 | . 2 ⊢ (𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊊ 𝐴) |
5 | imnan 624 | . 2 ⊢ ((𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊊ 𝐴) ↔ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴)) | |
6 | 4, 5 | mpbi 133 | 1 ⊢ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ 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: psstr 3049 |
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