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Mirrors > Home > ILE Home > Th. List > sseld | GIF version |
Description: Membership deduction from subclass relationship. (Contributed by NM, 15-Nov-1995.) |
Ref | Expression |
---|---|
sseld.1 | ⊢ (𝜑 → 𝐴 ⊆ 𝐵) |
Ref | Expression |
---|---|
sseld | ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseld.1 | . 2 ⊢ (𝜑 → 𝐴 ⊆ 𝐵) | |
2 | ssel 3096 | . 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) | |
3 | 1, 2 | syl 14 | 1 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1481 ⊆ wss 3076 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-11 1485 ax-4 1488 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 |
This theorem depends on definitions: df-bi 116 df-nf 1438 df-sb 1737 df-clab 2127 df-cleq 2133 df-clel 2136 df-in 3082 df-ss 3089 |
This theorem is referenced by: sselda 3102 sseldd 3103 ssneld 3104 elelpwi 3527 ssbrd 3979 uniopel 4186 onintonm 4441 sucprcreg 4472 ordsuc 4486 0elnn 4540 dmrnssfld 4810 nfunv 5164 opelf 5302 fvimacnv 5543 ffvelrn 5561 resflem 5592 f1imass 5683 dftpos3 6167 nnmordi 6420 mapsn 6592 ixpf 6622 diffifi 6796 ordiso2 6928 difinfinf 6994 prarloclemarch2 7251 ltexprlemrl 7442 cauappcvgprlemladdrl 7489 caucvgprlemladdrl 7510 caucvgprlem1 7511 axpre-suploclemres 7733 uzind 9186 supinfneg 9417 infsupneg 9418 ixxssxr 9713 elfz0add 9931 fzoss1 9979 frecuzrdgrclt 10219 fsum3cvg 11179 isumrpcl 11295 fproddccvg 11373 lmtopcnp 12458 txuni2 12464 tx1cn 12477 tx2cn 12478 txlm 12487 imasnopn 12507 xmetunirn 12566 mopnval 12650 metrest 12714 dedekindicc 12819 ivthdec 12830 limcimolemlt 12841 bj-nnord 13327 |
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