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Mirrors > Home > ILE Home > Th. List > sspsstrd | GIF version |
Description: Transitivity involving subclass and proper subclass inclusion. Deduction form of sspsstr 3044. (Contributed by David Moews, 1-May-2017.) |
Ref | Expression |
---|---|
sspsstrd.1 | ⊢ (φ → A ⊆ B) |
sspsstrd.2 | ⊢ (φ → B ⊊ 𝐶) |
Ref | Expression |
---|---|
sspsstrd | ⊢ (φ → A ⊊ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sspsstrd.1 | . 2 ⊢ (φ → A ⊆ B) | |
2 | sspsstrd.2 | . 2 ⊢ (φ → B ⊊ 𝐶) | |
3 | sspsstr 3044 | . 2 ⊢ ((A ⊆ B ∧ B ⊊ 𝐶) → A ⊊ 𝐶) | |
4 | 1, 2, 3 | syl2anc 391 | 1 ⊢ (φ → A ⊊ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 2911 ⊊ wpss 2912 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-in1 544 ax-in2 545 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-ne 2203 df-in 2918 df-ss 2925 df-pss 2927 |
This theorem is referenced by: (None) |
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