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Mirrors > Home > MPE Home > Th. List > pssirr | Structured version Visualization version 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 | pm3.24 402 | . 2 ⊢ ¬ (𝐴 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐴) | |
2 | dfpss3 4079 | . 2 ⊢ (𝐴 ⊊ 𝐴 ↔ (𝐴 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐴)) | |
3 | 1, 2 | mtbir 323 | 1 ⊢ ¬ 𝐴 ⊊ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 395 ⊆ wss 3941 ⊊ wpss 3942 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-tru 1536 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-v 3468 df-in 3948 df-ss 3958 df-pss 3960 |
This theorem is referenced by: porpss 7711 ltsopr 11024 |
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