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Mirrors > Home > MPE Home > Th. List > psstr | Structured version Visualization version GIF version |
Description: Transitive law for proper subclass. Theorem 9 of [Suppes] p. 23. (Contributed by NM, 7-Feb-1996.) |
Ref | Expression |
---|---|
psstr | ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | pssss 4069 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | pssss 4069 | . . 3 ⊢ (𝐵 ⊊ 𝐶 → 𝐵 ⊆ 𝐶) | |
3 | 1, 2 | sylan9ss 3977 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊆ 𝐶) |
4 | pssn2lp 4075 | . . . 4 ⊢ ¬ (𝐶 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) | |
5 | psseq1 4061 | . . . . 5 ⊢ (𝐴 = 𝐶 → (𝐴 ⊊ 𝐵 ↔ 𝐶 ⊊ 𝐵)) | |
6 | 5 | anbi1d 629 | . . . 4 ⊢ (𝐴 = 𝐶 → ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) ↔ (𝐶 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶))) |
7 | 4, 6 | mtbiri 328 | . . 3 ⊢ (𝐴 = 𝐶 → ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶)) |
8 | 7 | con2i 141 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → ¬ 𝐴 = 𝐶) |
9 | dfpss2 4059 | . 2 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶)) | |
10 | 3, 8, 9 | sylanbrc 583 | 1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 396 = wceq 1528 ⊆ wss 3933 ⊊ wpss 3934 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-clab 2797 df-cleq 2811 df-clel 2890 df-ne 3014 df-in 3940 df-ss 3949 df-pss 3951 |
This theorem is referenced by: sspsstr 4079 psssstr 4080 psstrd 4081 porpss 7442 inf3lem5 9083 ltsopr 10442 |
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