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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 4428 | . 2 ⊢ (𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ ∀𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥}𝐴 ⊆ 𝑦) | |
2 | sseq2 3590 | . . . 4 ⊢ (𝑥 = 𝑦 → (𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦)) | |
3 | 2 | elrab 3331 | . . 3 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} ↔ (𝑦 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑦)) |
4 | 3 | simprbi 479 | . 2 ⊢ (𝑦 ∈ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} → 𝐴 ⊆ 𝑦) |
5 | 1, 4 | mprgbir 2911 | 1 ⊢ 𝐴 ⊆ ∩ {𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 {crab 2900 ⊆ wss 3540 ∩ cint 4410 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rab 2905 df-v 3175 df-in 3547 df-ss 3554 df-int 4411 |
This theorem is referenced by: intmin 4432 wuncid 9444 mrcssid 16100 lspssid 18806 lbsextlem3 18981 aspssid 19154 sscls 20670 filufint 21534 spanss2 27588 shsval2i 27630 ococin 27651 chsupsn 27656 sssigagen 29535 dynkin 29557 igenss 33031 pclssidN 34199 dochocss 35673 rgspnssid 36759 |
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