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Mirrors > Home > MPE Home > Th. List > pssn2lp | Structured version Visualization version 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 3988 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) | |
2 | 1 | simprbi 497 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊆ 𝐴) |
3 | pssss 3997 | . . 3 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴) | |
4 | 2, 3 | nsyl 142 | . 2 ⊢ (𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊊ 𝐴) |
5 | imnan 400 | . 2 ⊢ ((𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊊ 𝐴) ↔ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴)) | |
6 | 4, 5 | mpbi 231 | 1 ⊢ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 ⊆ wss 3863 ⊊ wpss 3864 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-ext 2769 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-clab 2776 df-cleq 2788 df-clel 2863 df-ne 2985 df-in 3870 df-ss 3878 df-pss 3880 |
This theorem is referenced by: psstr 4006 cvnsym 29763 |
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