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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 3050 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | df-ral 2393 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | bitr4i 186 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 104 ∀wal 1310 ∈ wcel 1461 ∀wral 2388 ⊆ wss 3035 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-5 1404 ax-7 1405 ax-gen 1406 ax-ie1 1450 ax-ie2 1451 ax-8 1463 ax-11 1465 ax-4 1468 ax-17 1487 ax-i9 1491 ax-ial 1495 ax-i5r 1496 ax-ext 2095 |
This theorem depends on definitions: df-bi 116 df-nf 1418 df-sb 1717 df-clab 2100 df-cleq 2106 df-clel 2109 df-ral 2393 df-in 3041 df-ss 3048 |
This theorem is referenced by: ssrab 3139 eqsnm 3646 uni0b 3725 uni0c 3726 ssint 3751 ssiinf 3826 sspwuni 3861 dftr3 3988 tfis 4455 rninxp 4938 fnres 5195 eqfnfv3 5472 funimass3 5488 ffvresb 5535 tfrlemibxssdm 6176 tfr1onlembxssdm 6192 tfrcllembxssdm 6205 isbasis2g 12049 tgval2 12057 eltg2b 12060 tgss2 12085 basgen2 12087 bastop1 12089 unicld 12122 neipsm 12160 ssidcn 12215 bdss 12745 |
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