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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 4080. (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 4080 | . 2 ⊢ ((𝐴 ⊊ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊊ 𝐶) | |
4 | 1, 2, 3 | syl2anc 584 | 1 ⊢ (𝜑 → 𝐴 ⊊ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊆ 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: ackbij1lem15 9644 lsatssn0 36018 lsatexch 36059 lsatcvatlem 36065 lkrpssN 36179 |
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