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Mirrors > Home > MPE Home > Th. List > ssintub | Structured version Visualization version GIF version |
Description: Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000.) |
Ref | Expression |
---|---|
ssintub | ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ssint 4968 | . 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}𝐴 ⊆ 𝑦) | |
2 | sseq2 4021 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦)) | |
3 | 2 | elrab 3694 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ (𝑦 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑦)) |
4 | 3 | simprbi 496 | . 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → 𝐴 ⊆ 𝑦) |
5 | 1, 4 | mprgbir 3065 | 1 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2105 {crab 3432 ⊆ wss 3962 ∩ cint 4950 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-11 2154 ax-ext 2705 |
This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1539 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ral 3059 df-rab 3433 df-v 3479 df-ss 3979 df-int 4951 |
This theorem is referenced by: intmin 4972 cofon2 8709 naddunif 8729 wuncid 10780 mrcssid 17661 rgspnssid 20630 lspssid 21000 lbsextlem3 21179 aspssid 21915 sscls 23079 filufint 23943 spanss2 31373 shsval2i 31415 ococin 31436 chsupsn 31441 fldgenssid 33294 sssigagen 34125 dynkin 34147 igenss 38048 pclssidN 39877 dochocss 41348 intubeu 48772 |
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