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Mirrors > Home > MPE Home > Th. List > psssstrd | Structured version Visualization version GIF version |
Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of psssstr 4085. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
psssstrd.1 | ⊢ (𝜑 → 𝐴 ⊊ 𝐵) |
psssstrd.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
psssstrd | ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psssstrd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊊ 𝐵) | |
2 | psssstrd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
3 | psssstr 4085 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶) | |
4 | 1, 2, 3 | syl2anc 586 | 1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ wss 3938 ⊊ wpss 3939 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2802 df-cleq 2816 df-clel 2895 df-ne 3019 df-in 3945 df-ss 3954 df-pss 3956 |
This theorem is referenced by: ackbij1lem15 9658 lsatssn0 36140 lsatexch 36181 lsatcvatlem 36187 lkrpssN 36301 |
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