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Mirrors > Home > MPE Home > Th. List > Mathboxes > atex | Structured version Visualization version GIF version |
Description: At least one atom exists. (Contributed by NM, 15-Jul-2012.) |
Ref | Expression |
---|---|
atex.1 | ⊢ 𝐴 = (Atoms‘𝐾) |
Ref | Expression |
---|---|
atex | ⊢ (𝐾 ∈ HL → 𝐴 ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | atex.1 | . . . 4 ⊢ 𝐴 = (Atoms‘𝐾) | |
2 | 1 | hl2at 36023 | . . 3 ⊢ (𝐾 ∈ HL → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞) |
3 | df-rex 3087 | . . . 4 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞 ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞)) | |
4 | exsimpl 1832 | . . . 4 ⊢ (∃𝑝(𝑝 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞) → ∃𝑝 𝑝 ∈ 𝐴) | |
5 | 3, 4 | sylbi 209 | . . 3 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞 → ∃𝑝 𝑝 ∈ 𝐴) |
6 | 2, 5 | syl 17 | . 2 ⊢ (𝐾 ∈ HL → ∃𝑝 𝑝 ∈ 𝐴) |
7 | n0 4190 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑝 𝑝 ∈ 𝐴) | |
8 | 6, 7 | sylibr 226 | 1 ⊢ (𝐾 ∈ HL → 𝐴 ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 387 = wceq 1508 ∃wex 1743 ∈ wcel 2051 ≠ wne 2960 ∃wrex 3082 ∅c0 4172 ‘cfv 6185 Atomscatm 35881 HLchlt 35968 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1759 ax-4 1773 ax-5 1870 ax-6 1929 ax-7 1966 ax-8 2053 ax-9 2060 ax-10 2080 ax-11 2094 ax-12 2107 ax-13 2302 ax-ext 2743 ax-rep 5045 ax-sep 5056 ax-nul 5063 ax-pow 5115 ax-pr 5182 ax-un 7277 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 835 df-3an 1071 df-tru 1511 df-ex 1744 df-nf 1748 df-sb 2017 df-mo 2548 df-eu 2585 df-clab 2752 df-cleq 2764 df-clel 2839 df-nfc 2911 df-ne 2961 df-ral 3086 df-rex 3087 df-reu 3088 df-rab 3090 df-v 3410 df-sbc 3675 df-csb 3780 df-dif 3825 df-un 3827 df-in 3829 df-ss 3836 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4709 df-iun 4790 df-br 4926 df-opab 4988 df-mpt 5005 df-id 5308 df-xp 5409 df-rel 5410 df-cnv 5411 df-co 5412 df-dm 5413 df-rn 5414 df-res 5415 df-ima 5416 df-iota 6149 df-fun 6187 df-fn 6188 df-f 6189 df-f1 6190 df-fo 6191 df-f1o 6192 df-fv 6193 df-riota 6935 df-ov 6977 df-oprab 6978 df-proset 17408 df-poset 17426 df-plt 17438 df-lub 17454 df-glb 17455 df-join 17456 df-meet 17457 df-p0 17519 df-p1 17520 df-lat 17526 df-clat 17588 df-oposet 35794 df-ol 35796 df-oml 35797 df-covers 35884 df-ats 35885 df-atl 35916 df-cvlat 35940 df-hlat 35969 |
This theorem is referenced by: llnn0 36134 lplnn0N 36165 lvoln0N 36209 |
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