![]() |
Mathbox for Norm Megill |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > Mathboxes > atcvr0 | Structured version Visualization version GIF version |
Description: An atom covers zero. (atcv0 32145 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 2727 | . . 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 38747 | . 2 ⊢ (𝐾 ∈ 𝐷 → (𝑃 ∈ 𝐴 ↔ (𝑃 ∈ (Base‘𝐾) ∧ 0 𝐶𝑃))) |
6 | 5 | simplbda 499 | 1 ⊢ ((𝐾 ∈ 𝐷 ∧ 𝑃 ∈ 𝐴) → 0 𝐶𝑃) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 ∈ wcel 2099 class class class wbr 5142 ‘cfv 6542 Basecbs 17173 0.cp0 18408 ⋖ ccvr 38723 Atomscatm 38724 |
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 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-iota 6494 df-fun 6544 df-fv 6550 df-ats 38728 |
This theorem is referenced by: 0ltat 38752 leatb 38753 atnle0 38770 atlen0 38771 atcmp 38772 atcvreq0 38775 atcvr0eq 38888 lnnat 38889 athgt 38918 ps-2 38940 lhp0lt 39465 |
Copyright terms: Public domain | W3C validator |