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Mirrors > Home > ILE Home > Th. List > ssel2 | GIF version |
Description: Membership relationships follow from a subclass relationship. (Contributed by NM, 7-Jun-2004.) |
Ref | Expression |
---|---|
ssel2 | ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssel 3091 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) | |
2 | 1 | imp 123 | 1 ⊢ ((𝐴 ⊆ 𝐵 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 ⊆ 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: elnn 4519 funimass4 5472 fvelimab 5477 ssimaex 5482 funconstss 5538 rexima 5656 ralima 5657 1st2nd 6079 f1o2ndf1 6125 tfri1dALT 6248 eldju1st 6956 axsuploc 7837 lbinf 8706 dfinfre 8714 lbzbi 9408 elfzom1elp1fzo 9979 ssfzo12 10001 seq3split 10252 shftlem 10588 tgcl 12233 neipsm 12323 txbasval 12436 elmopn2 12618 metrest 12675 cncfmet 12748 negcncf 12757 |
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