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Mirrors > Home > MPE Home > Th. List > 0ellim | Structured version Visualization version GIF version |
Description: A limit ordinal contains the empty set. (Contributed by NM, 15-May-1994.) |
Ref | Expression |
---|---|
0ellim | ⊢ (Lim 𝐴 → ∅ ∈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nlim0 6249 | . . . 4 ⊢ ¬ Lim ∅ | |
2 | limeq 6203 | . . . 4 ⊢ (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅)) | |
3 | 1, 2 | mtbiri 330 | . . 3 ⊢ (𝐴 = ∅ → ¬ Lim 𝐴) |
4 | 3 | necon2ai 2961 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ ∅) |
5 | limord 6250 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
6 | ord0eln0 6245 | . . 3 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (Lim 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
8 | 4, 7 | mpbird 260 | 1 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 = wceq 1543 ∈ wcel 2112 ≠ wne 2932 ∅c0 4223 Ord word 6190 Lim wlim 6192 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-11 2160 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2073 df-clab 2715 df-cleq 2728 df-clel 2809 df-ne 2933 df-ral 3056 df-rex 3057 df-rab 3060 df-v 3400 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-op 4534 df-uni 4806 df-br 5040 df-opab 5102 df-tr 5147 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-ord 6194 df-lim 6196 |
This theorem is referenced by: limuni3 7609 peano1 7645 oe1m 8251 oalimcl 8266 oaass 8267 oarec 8268 omlimcl 8284 odi 8285 oen0 8292 oewordri 8298 oelim2 8301 oeoalem 8302 oeoelem 8304 limensuci 8800 rankxplim2 9461 rankxplim3 9462 r1limwun 10315 |
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