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Mirrors > Home > MPE Home > Th. List > Mathboxes > atssbase | Structured version Visualization version GIF version |
Description: The set of atoms is a subset of the base set. (atssch 32273 analog.) (Contributed by NM, 21-Oct-2011.) |
Ref | Expression |
---|---|
atombase.b | ⊢ 𝐵 = (Base‘𝐾) |
atombase.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atssbase | ⊢ 𝐴 ⊆ 𝐵 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atombase.b | . . 3 ⊢ 𝐵 = (Base‘𝐾) | |
2 | atombase.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
3 | 1, 2 | atbase 39000 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) |
4 | 3 | ssriv 3982 | 1 ⊢ 𝐴 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1534 ⊆ wss 3946 ‘cfv 6546 Basecbs 17208 Atomscatm 38974 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2697 ax-sep 5296 ax-nul 5303 ax-pr 5425 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2704 df-cleq 2718 df-clel 2803 df-nfc 2878 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-nul 4323 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4906 df-br 5146 df-opab 5208 df-mpt 5229 df-id 5572 df-xp 5680 df-rel 5681 df-cnv 5682 df-co 5683 df-dm 5684 df-iota 6498 df-fun 6548 df-fv 6554 df-ats 38978 |
This theorem is referenced by: atlatmstc 39030 atlatle 39031 pmapssbaN 39472 pmaple 39473 polsubN 39619 2polvalN 39626 2polssN 39627 3polN 39628 2pmaplubN 39638 paddunN 39639 poldmj1N 39640 pnonsingN 39645 ispsubcl2N 39659 psubclinN 39660 paddatclN 39661 polsubclN 39664 poml4N 39665 |
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