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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 4178 | . . 3 ⊢ ¬ ∅ ∈ ∅ | |
2 | simp2 1117 | . . 3 ⊢ ((Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅) → ∅ ∈ ∅) | |
3 | 1, 2 | mto 189 | . 2 ⊢ ¬ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅) |
4 | dflim2 6079 | . 2 ⊢ (Lim ∅ ↔ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅)) | |
5 | 3, 4 | mtbir 315 | 1 ⊢ ¬ Lim ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ w3a 1068 = wceq 1507 ∈ wcel 2048 ∅c0 4173 ∪ cuni 4706 Ord word 6022 Lim wlim 6024 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2745 ax-sep 5054 ax-nul 5061 ax-pr 5180 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2754 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3678 df-dif 3828 df-un 3830 df-in 3832 df-ss 3839 df-pss 3841 df-nul 4174 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-tr 5025 df-eprel 5310 df-po 5319 df-so 5320 df-fr 5359 df-we 5361 df-ord 6026 df-lim 6028 |
This theorem is referenced by: 0ellim 6085 tz7.44lem1 7838 tz7.44-3 7841 cflim2 9475 rankcf 9989 dfrdg4 32873 limsucncmpi 33253 |
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