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Mirrors > Home > ILE Home > Th. List > dfss3 | GIF version |
Description: Alternate definition of subclass relationship. (Contributed by NM, 14-Oct-1999.) |
Ref | Expression |
---|---|
dfss3 | ⊢ (A ⊆ B ↔ ∀x ∈ A x ∈ B) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfss2 2928 | . 2 ⊢ (A ⊆ B ↔ ∀x(x ∈ A → x ∈ B)) | |
2 | df-ral 2305 | . 2 ⊢ (∀x ∈ A x ∈ B ↔ ∀x(x ∈ A → x ∈ B)) | |
3 | 1, 2 | bitr4i 176 | 1 ⊢ (A ⊆ B ↔ ∀x ∈ A x ∈ B) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∀wal 1240 ∈ wcel 1390 ∀wral 2300 ⊆ wss 2911 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-11 1394 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-ral 2305 df-in 2918 df-ss 2925 |
This theorem is referenced by: ssrab 3012 eqsnm 3517 uni0b 3596 uni0c 3597 ssint 3622 ssiinf 3697 sspwuni 3730 dftr3 3849 tfis 4249 rninxp 4707 fnres 4958 eqfnfv3 5210 funimass3 5226 ffvresb 5271 tfrlemibxssdm 5882 bdss 9319 |
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