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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 30122 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 36427 | . 2 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵) |
4 | 3 | ssriv 3973 | 1 ⊢ 𝐴 ⊆ 𝐵 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ⊆ wss 3938 ‘cfv 6357 Basecbs 16485 Atomscatm 36401 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-mpt 5149 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-iota 6316 df-fun 6359 df-fv 6365 df-ats 36405 |
This theorem is referenced by: atlatmstc 36457 atlatle 36458 pmapssbaN 36898 pmaple 36899 polsubN 37045 2polvalN 37052 2polssN 37053 3polN 37054 2pmaplubN 37064 paddunN 37065 poldmj1N 37066 pnonsingN 37071 ispsubcl2N 37085 psubclinN 37086 paddatclN 37087 polsubclN 37090 poml4N 37091 |
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