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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 3056 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | df-ral 2398 | . 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 1314 ∈ wcel 1465 ∀wral 2393 ⊆ wss 3041 |
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 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-11 1469 ax-4 1472 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 |
This theorem depends on definitions: df-bi 116 df-nf 1422 df-sb 1721 df-clab 2104 df-cleq 2110 df-clel 2113 df-ral 2398 df-in 3047 df-ss 3054 |
This theorem is referenced by: ssrab 3145 eqsnm 3652 uni0b 3731 uni0c 3732 ssint 3757 ssiinf 3832 sspwuni 3867 dftr3 4000 tfis 4467 rninxp 4952 fnres 5209 eqfnfv3 5488 funimass3 5504 ffvresb 5551 tfrlemibxssdm 6192 tfr1onlembxssdm 6208 tfrcllembxssdm 6221 suplocsr 7585 isbasis2g 12139 tgval2 12147 eltg2b 12150 tgss2 12175 basgen2 12177 bastop1 12179 unicld 12212 neipsm 12250 ssidcn 12306 bdss 12989 |
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