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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 3075 | . 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐵 ⊆ 𝐶) → 𝐴 ⊆ 𝐶) | |
4 | 1, 2, 3 | syl2anc 408 | 1 ⊢ (𝜑 → 𝐴 ⊆ 𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ⊆ wss 3041 |
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 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-in 3047 df-ss 3054 |
This theorem is referenced by: sstrid 3078 sstrdi 3079 ssdif2d 3185 tfisi 4471 funss 5112 fssxp 5260 fvmptssdm 5473 suppssfv 5946 suppssov1 5947 tposss 6111 tfrlem1 6173 tfrlemibfn 6193 tfr1onlembfn 6209 tfr1onlemubacc 6211 tfr1onlemres 6214 tfrcllembfn 6222 tfrcllemubacc 6224 tfrcllemres 6227 ecinxp 6472 undifdc 6780 sbthlem1 6813 iseqf1olemnab 10229 isumss 11128 ennnfoneleminc 11851 strsetsid 11919 strleund 11974 ntrss 12215 neiint 12241 neiss 12246 restopnb 12277 iscnp4 12314 blssps 12523 blss 12524 xmettx 12606 tgqioo 12643 rescncf 12664 suplociccreex 12698 suplociccex 12699 dvbss 12750 dvbsssg 12751 dvfgg 12753 dvcnp2cntop 12759 dvcn 12760 dvaddxxbr 12761 dvmulxxbr 12762 dvcoapbr 12767 |
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