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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 4964 | . 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}𝐴 ⊆ 𝑦) | |
| 2 | sseq2 4010 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦)) | |
| 3 | 2 | elrab 3692 | . . 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 3068 | 1 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} |
| Colors of variables: wff setvar class |
| Syntax hints: ∈ wcel 2108 {crab 3436 ⊆ wss 3951 ∩ cint 4946 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-11 2157 ax-ext 2708 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-tru 1543 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rab 3437 df-v 3482 df-ss 3968 df-int 4947 |
| This theorem is referenced by: intmin 4968 cofon2 8711 naddunif 8731 wuncid 10783 mrcssid 17660 rgspnssid 20614 lspssid 20983 lbsextlem3 21162 aspssid 21898 sscls 23064 filufint 23928 spanss2 31364 shsval2i 31406 ococin 31427 chsupsn 31432 fldgenssid 33315 sssigagen 34146 dynkin 34168 igenss 38069 pclssidN 39897 dochocss 41368 intubeu 48873 |
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