| Mathbox for Norm Megill |
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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 39407 | . . 3 ⊢ (𝐾 ∈ HL → ∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞) |
| 3 | df-rex 3071 | . . . 4 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞 ↔ ∃𝑝(𝑝 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞)) | |
| 4 | exsimpl 1868 | . . . 4 ⊢ (∃𝑝(𝑝 ∈ 𝐴 ∧ ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞) → ∃𝑝 𝑝 ∈ 𝐴) | |
| 5 | 3, 4 | sylbi 217 | . . 3 ⊢ (∃𝑝 ∈ 𝐴 ∃𝑞 ∈ 𝐴 𝑝 ≠ 𝑞 → ∃𝑝 𝑝 ∈ 𝐴) |
| 6 | 2, 5 | syl 17 | . 2 ⊢ (𝐾 ∈ HL → ∃𝑝 𝑝 ∈ 𝐴) |
| 7 | n0 4353 | . 2 ⊢ (𝐴 ≠ ∅ ↔ ∃𝑝 𝑝 ∈ 𝐴) | |
| 8 | 6, 7 | sylibr 234 | 1 ⊢ (𝐾 ∈ HL → 𝐴 ≠ ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2940 ∃wrex 3070 ∅c0 4333 ‘cfv 6561 Atomscatm 39264 HLchlt 39351 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-riota 7388 df-ov 7434 df-oprab 7435 df-proset 18340 df-poset 18359 df-plt 18375 df-lub 18391 df-glb 18392 df-join 18393 df-meet 18394 df-p0 18470 df-p1 18471 df-lat 18477 df-clat 18544 df-oposet 39177 df-ol 39179 df-oml 39180 df-covers 39267 df-ats 39268 df-atl 39299 df-cvlat 39323 df-hlat 39352 |
| This theorem is referenced by: llnn0 39518 lplnn0N 39549 lvoln0N 39593 |
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