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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 3146 | . 2 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
2 | df-ral 2460 | . 2 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵)) | |
3 | 1, 2 | bitr4i 187 | 1 ⊢ (𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 105 ∀wal 1351 ∈ wcel 2148 ∀wral 2455 ⊆ wss 3131 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-11 1506 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-ext 2159 |
This theorem depends on definitions: df-bi 117 df-nf 1461 df-sb 1763 df-clab 2164 df-cleq 2170 df-clel 2173 df-ral 2460 df-in 3137 df-ss 3144 |
This theorem is referenced by: ssrab 3235 eqsnm 3757 uni0b 3836 uni0c 3837 ssint 3862 ssiinf 3938 sspwuni 3973 dftr3 4107 tfis 4584 rninxp 5074 fnres 5334 eqfnfv3 5618 funimass3 5635 ffvresb 5682 tfrlemibxssdm 6331 tfr1onlembxssdm 6347 tfrcllembxssdm 6360 exmidontriimlem3 7225 suplocsr 7811 imasaddfnlemg 12741 isbasis2g 13706 tgval2 13712 eltg2b 13715 tgss2 13740 basgen2 13742 bastop1 13744 unicld 13777 neipsm 13815 ssidcn 13871 bdss 14777 |
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