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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 4023 | . . 3 ⊢ (𝐴 ⊊ 𝐵 → 𝐴 ⊆ 𝐵) | |
2 | pssss 4023 | . . 3 ⊢ (𝐵 ⊊ 𝐶 → 𝐵 ⊆ 𝐶) | |
3 | 1, 2 | sylan9ss 3928 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊆ 𝐶) |
4 | pssn2lp 4029 | . . . 4 ⊢ ¬ (𝐶 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) | |
5 | psseq1 4015 | . . . . 5 ⊢ (𝐴 = 𝐶 → (𝐴 ⊊ 𝐵 ↔ 𝐶 ⊊ 𝐵)) | |
6 | 5 | anbi1d 632 | . . . 4 ⊢ (𝐴 = 𝐶 → ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) ↔ (𝐶 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶))) |
7 | 4, 6 | mtbiri 330 | . . 3 ⊢ (𝐴 = 𝐶 → ¬ (𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶)) |
8 | 7 | con2i 141 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → ¬ 𝐴 = 𝐶) |
9 | dfpss2 4013 | . 2 ⊢ (𝐴 ⊊ 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ ¬ 𝐴 = 𝐶)) | |
10 | 3, 8, 9 | sylanbrc 586 | 1 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊊ 𝐶) → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 399 = wceq 1538 ⊆ wss 3881 ⊊ wpss 3882 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ne 2988 df-v 3443 df-in 3888 df-ss 3898 df-pss 3900 |
This theorem is referenced by: sspsstr 4033 psssstr 4034 psstrd 4035 porpss 7433 inf3lem5 9079 ltsopr 10443 |
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