![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4047 | . . . 4 ⊢ (𝐴 ⊊ 𝐵 ↔ (𝐴 ⊆ 𝐵 ∧ ¬ 𝐵 ⊆ 𝐴)) | |
2 | 1 | simprbi 498 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊆ 𝐴) |
3 | pssss 4056 | . . 3 ⊢ (𝐵 ⊊ 𝐴 → 𝐵 ⊆ 𝐴) | |
4 | 2, 3 | nsyl 140 | . 2 ⊢ (𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊊ 𝐴) |
5 | imnan 401 | . 2 ⊢ ((𝐴 ⊊ 𝐵 → ¬ 𝐵 ⊊ 𝐴) ↔ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴)) | |
6 | 4, 5 | mpbi 229 | 1 ⊢ ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ⊆ wss 3911 ⊊ wpss 3912 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-v 3446 df-in 3918 df-ss 3928 df-pss 3930 |
This theorem is referenced by: psstr 4065 cvnsym 31274 |
Copyright terms: Public domain | W3C validator |