![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6435 | . . . 4 ⊢ ¬ Lim ∅ | |
2 | limeq 6388 | . . . 4 ⊢ (𝐴 = ∅ → (Lim 𝐴 ↔ Lim ∅)) | |
3 | 1, 2 | mtbiri 326 | . . 3 ⊢ (𝐴 = ∅ → ¬ Lim 𝐴) |
4 | 3 | necon2ai 2960 | . 2 ⊢ (Lim 𝐴 → 𝐴 ≠ ∅) |
5 | limord 6436 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
6 | ord0eln0 6431 | . . 3 ⊢ (Ord 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) | |
7 | 5, 6 | syl 17 | . 2 ⊢ (Lim 𝐴 → (∅ ∈ 𝐴 ↔ 𝐴 ≠ ∅)) |
8 | 4, 7 | mpbird 256 | 1 ⊢ (Lim 𝐴 → ∅ ∈ 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 = wceq 1534 ∈ wcel 2099 ≠ wne 2930 ∅c0 4325 Ord word 6375 Lim wlim 6377 |
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-ext 2697 ax-sep 5304 ax-nul 5311 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 df-3an 1086 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-clab 2704 df-cleq 2718 df-clel 2803 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3464 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3967 df-nul 4326 df-if 4534 df-pw 4609 df-sn 4634 df-pr 4636 df-op 4640 df-uni 4914 df-br 5154 df-opab 5216 df-tr 5271 df-eprel 5586 df-po 5594 df-so 5595 df-fr 5637 df-we 5639 df-ord 6379 df-lim 6381 |
This theorem is referenced by: limuni3 7862 peano1 7900 peano1OLD 7901 oe1m 8575 oalimcl 8590 oaass 8591 oarec 8592 omlimcl 8608 odi 8609 oen0 8616 oewordri 8622 oelim2 8625 oeoalem 8626 oeoelem 8628 limensuci 9191 rankxplim2 9923 rankxplim3 9924 r1limwun 10779 constr01 33600 omlimcl2 42907 oe0suclim 42943 |
Copyright terms: Public domain | W3C validator |