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Mathbox for Norm Megill |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > atcvr0 | Structured version Visualization version GIF version |
Description: An atom covers zero. (atcv0 30125 analog.) (Contributed by NM, 4-Nov-2011.) |
Ref | Expression |
---|---|
atomcvr0.z | ⊢ 0 = (0.‘𝐾) |
atomcvr0.c | ⊢ 𝐶 = ( ⋖ ‘𝐾) |
atomcvr0.a | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atcvr0 | ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐴) → 0 𝐶𝑃) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2798 | . . 3 ⊢ (Base‘𝐾) = (Base‘𝐾) | |
2 | atomcvr0.z | . . 3 ⊢ 0 = (0.‘𝐾) | |
3 | atomcvr0.c | . . 3 ⊢ 𝐶 = ( ⋖ ‘𝐾) | |
4 | atomcvr0.a | . . 3 ⊢ 𝐴 = (Atoms‘𝐾) | |
5 | 1, 2, 3, 4 | isat 36582 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ (Base‘𝐾) ∧ 0 𝐶𝑃))) |
6 | 5 | simplbda 503 | 1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐴) → 0 𝐶𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1538 ∈ wcel 2111 class class class wbr 5030 ‘cfv 6324 Basecbs 16475 0.cp0 17639 ⋖ ccvr 36558 Atomscatm 36559 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-ats 36563 |
This theorem is referenced by: 0ltat 36587 leatb 36588 atnle0 36605 atlen0 36606 atcmp 36607 atcvreq0 36610 atcvr0eq 36722 lnnat 36723 athgt 36752 ps-2 36774 lhp0lt 37299 |
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