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Mirrors > Home > MPE Home > Th. List > nlim0 | Structured version Visualization version GIF version |
Description: The empty set is not a limit ordinal. (Contributed by NM, 24-Mar-1995.) (Proof shortened by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
nlim0 | ⊢ ¬ Lim ∅ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noel 4298 | . . 3 ⊢ ¬ ∅ ∈ ∅ | |
2 | simp2 1133 | . . 3 ⊢ ((Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅) → ∅ ∈ ∅) | |
3 | 1, 2 | mto 199 | . 2 ⊢ ¬ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅) |
4 | dflim2 6249 | . 2 ⊢ (Lim ∅ ↔ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅)) | |
5 | 3, 4 | mtbir 325 | 1 ⊢ ¬ Lim ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∅c0 4293 ∪ cuni 4840 Ord word 6192 Lim wlim 6194 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-tr 5175 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-ord 6196 df-lim 6198 |
This theorem is referenced by: 0ellim 6255 tz7.44lem1 8043 tz7.44-3 8046 cflim2 9687 rankcf 10201 dfrdg4 33414 limsucncmpi 33795 |
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