![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > sstrd | GIF version |
Description: Subclass transitivity deduction. (Contributed by NM, 2-Jun-2004.) |
Ref | Expression |
---|---|
sstrd.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
sstrd.2 | ⊢ (𝜑 → 𝐵 ⊆ 𝐶) |
Ref | Expression |
---|---|
sstrd | ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sstrd.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | sstrd.2 | . 2 ⊢ (𝜑 → 𝐵 ⊆ 𝐶) | |
3 | sstr 3033 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) | |
4 | 1, 2, 3 | syl2anc 403 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 2999 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-5 1381 ax-7 1382 ax-gen 1383 ax-ie1 1427 ax-ie2 1428 ax-8 1440 ax-11 1442 ax-4 1445 ax-17 1464 ax-i9 1468 ax-ial 1472 ax-i5r 1473 ax-ext 2070 |
This theorem depends on definitions: df-bi 115 df-nf 1395 df-sb 1693 df-clab 2075 df-cleq 2081 df-clel 2084 df-in 3005 df-ss 3012 |
This theorem is referenced by: syl5ss 3036 syl6ss 3037 ssdif2d 3139 tfisi 4402 funss 5034 fssxp 5178 fvmptssdm 5387 suppssfv 5852 suppssov1 5853 tposss 6011 tfrlem1 6073 tfrlemibfn 6093 tfr1onlembfn 6109 tfr1onlemubacc 6111 tfr1onlemres 6114 tfrcllembfn 6122 tfrcllemubacc 6124 tfrcllemres 6127 ecinxp 6367 undifdc 6634 sbthlem1 6666 iseqf1olemnab 9917 isumss 10783 strsetsid 11527 strleund 11581 rescncf 11637 |
Copyright terms: Public domain | W3C validator |