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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 | pm3.24 926 | . 2 ⊢ ¬ (𝐴 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐴) | |
2 | dfpss3 3693 | . 2 ⊢ (𝐴 ⊊ 𝐴 ↔ (𝐴 ⊆ 𝐴 ∧ ¬ 𝐴 ⊆ 𝐴)) | |
3 | 1, 2 | mtbir 313 | 1 ⊢ ¬ 𝐴 ⊊ 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ wa 384 ⊆ wss 3574 ⊊ wpss 3575 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-ne 2795 df-in 3581 df-ss 3588 df-pss 3590 |
This theorem is referenced by: porpss 6941 ltsopr 9854 |
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