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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 6373 | . . . 4 ⊢ ¬ Lim ∅ | |
2 | limeq 6326 | . . . 4 ⊢ (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅)) | |
3 | 1, 2 | mtbiri 327 | . . 3 ⊢ (𝐴 = ∅ → ¬ Lim 𝐴) |
4 | 3 | necon2ai 2972 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ ∅) |
5 | limord 6374 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
6 | ord0eln0 6369 | . . 3 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (Lim 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
8 | 4, 7 | mpbird 257 | 1 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1542 ∈ wcel 2107 ≠ wne 2942 ∅c0 4281 Ord word 6313 Lim wlim 6315 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2709 ax-sep 5255 ax-nul 5262 ax-pr 5383 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2716 df-cleq 2730 df-clel 2816 df-ne 2943 df-ral 3064 df-rex 3073 df-rab 3407 df-v 3446 df-dif 3912 df-un 3914 df-in 3916 df-ss 3926 df-pss 3928 df-nul 4282 df-if 4486 df-pw 4561 df-sn 4586 df-pr 4588 df-op 4592 df-uni 4865 df-br 5105 df-opab 5167 df-tr 5222 df-eprel 5535 df-po 5543 df-so 5544 df-fr 5586 df-we 5588 df-ord 6317 df-lim 6319 |
This theorem is referenced by: limuni3 7779 peano1 7816 peano1OLD 7817 oe1m 8460 oalimcl 8475 oaass 8476 oarec 8477 omlimcl 8493 odi 8494 oen0 8501 oewordri 8507 oelim2 8510 oeoalem 8511 oeoelem 8513 limensuci 9031 rankxplim2 9750 rankxplim3 9751 r1limwun 10606 omlimcl2 41453 |
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