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Mirrors > Home > ILE Home > Th. List > syl5sseq | GIF version |
Description: Subclass transitivity deduction. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) |
Ref | Expression |
---|---|
syl5sseq.1 | ⊢ 𝐵 ⊆ 𝐴 |
syl5sseq.2 | ⊢ (𝜑 → 𝐴 = 𝐶) |
Ref | Expression |
---|---|
syl5sseq | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | syl5sseq.2 | . 2 ⊢ (𝜑 → 𝐴 = 𝐶) | |
2 | syl5sseq.1 | . 2 ⊢ 𝐵 ⊆ 𝐴 | |
3 | sseq2 2967 | . . 3 ⊢ (𝐴 = 𝐶 → (𝐵 ⊆ 𝐴 ↔ 𝐵 ⊆ 𝐶)) | |
4 | 3 | biimpa 280 | . 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 ⊆ 𝐴) → 𝐵 ⊆ 𝐶) |
5 | 1, 2, 4 | sylancl 392 | 1 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1243 ⊆ wss 2917 |
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-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-11 1397 ax-4 1400 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 |
This theorem depends on definitions: df-bi 110 df-nf 1350 df-sb 1646 df-clab 2027 df-cleq 2033 df-clel 2036 df-in 2924 df-ss 2931 |
This theorem is referenced by: fndmdif 5272 fneqeql2 5276 fconst4m 5381 f1opw2 5706 ecss 6147 fopwdom 6310 phplem2 6316 nn0supp 8234 monoord2 9236 |
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