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Mirrors > Home > NFE Home > Th. List > ssintub | 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 3943 | . 2 ⊢ (A ⊆ ∩{x ∈ B ∣ A ⊆ x} ↔ ∀y ∈ {x ∈ B ∣ A ⊆ x}A ⊆ y) | |
2 | sseq2 3294 | . . . 4 ⊢ (x = y → (A ⊆ x ↔ A ⊆ y)) | |
3 | 2 | elrab 2995 | . . 3 ⊢ (y ∈ {x ∈ B ∣ A ⊆ x} ↔ (y ∈ B ∧ A ⊆ y)) |
4 | 3 | simprbi 450 | . 2 ⊢ (y ∈ {x ∈ B ∣ A ⊆ x} → A ⊆ y) |
5 | 1, 4 | mprgbir 2685 | 1 ⊢ A ⊆ ∩{x ∈ B ∣ A ⊆ x} |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1710 {crab 2619 ⊆ wss 3258 ∩cint 3927 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2479 df-ral 2620 df-rab 2624 df-v 2862 df-nin 3212 df-compl 3213 df-in 3214 df-ss 3260 df-int 3928 |
This theorem is referenced by: intmin 3947 |
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