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Mirrors > Home > ILE Home > Th. List > sstrid | GIF version |
Description: Subclass transitivity deduction. (Contributed by NM, 6-Feb-2014.) |
Ref | Expression |
---|---|
sstrid.1 | ⊢ 𝐴 ⊆ 𝐵 |
sstrid.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
sstrid | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstrid.1 | . . 3 ⊢ 𝐴 ⊆ 𝐵 | |
2 | 1 | a1i 9 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
3 | sstrid.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
4 | 2, 3 | sstrd 3107 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3071 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-11 1484 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 |
This theorem depends on definitions: df-bi 116 df-nf 1437 df-sb 1736 df-clab 2126 df-cleq 2132 df-clel 2135 df-in 3077 df-ss 3084 |
This theorem is referenced by: cossxp2 5062 fimacnv 5549 smores2 6191 f1imaen2g 6687 phplem4dom 6756 isinfinf 6791 fidcenumlemrk 6842 casef 6973 genipv 7320 fzossnn0 9955 seq3split 10255 ctinf 11946 tgcl 12236 epttop 12262 ntrin 12296 cnconst2 12405 cnrest2 12408 cnptopresti 12410 cnptoprest2 12412 hmeores 12487 blin2 12604 ivthdec 12794 limcdifap 12803 limcresi 12807 dvfgg 12829 dvcnp2cntop 12835 dvaddxxbr 12837 |
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