Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
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 3141 | . 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 2141 ⊆ wss 3121 |
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 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-11 1499 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-ext 2152 |
This theorem depends on definitions: df-bi 116 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-in 3127 df-ss 3134 |
This theorem is referenced by: elnn 4590 funimass4 5547 fvelimab 5552 ssimaex 5557 funconstss 5614 rexima 5734 ralima 5735 1st2nd 6160 f1o2ndf1 6207 tfri1dALT 6330 eldju1st 7048 axsuploc 7992 lbinf 8864 dfinfre 8872 lbzbi 9575 elfzom1elp1fzo 10158 ssfzo12 10180 seq3split 10435 shftlem 10780 uzwodc 11992 tgcl 12858 neipsm 12948 txbasval 13061 elmopn2 13243 metrest 13300 cncfmet 13373 negcncf 13382 reeff1olem 13486 |
Copyright terms: Public domain | W3C validator |