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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 4299 | . . 3 ⊢ ¬ ∅ ∈ ∅ | |
2 | simp2 1133 | . . 3 ⊢ ((Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅) → ∅ ∈ ∅) | |
3 | 1, 2 | mto 199 | . 2 ⊢ ¬ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅) |
4 | dflim2 6250 | . 2 ⊢ (Lim ∅ ↔ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅)) | |
5 | 3, 4 | mtbir 325 | 1 ⊢ ¬ Lim ∅ |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ∧ w3a 1083 = wceq 1536 ∈ wcel 2113 ∅c0 4294 ∪ cuni 4841 Ord word 6193 Lim wlim 6195 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-sep 5206 ax-nul 5213 ax-pr 5333 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-rab 3150 df-v 3499 df-sbc 3776 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-op 4577 df-uni 4842 df-br 5070 df-opab 5132 df-tr 5176 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-ord 6197 df-lim 6199 |
This theorem is referenced by: 0ellim 6256 tz7.44lem1 8044 tz7.44-3 8047 cflim2 9688 rankcf 10202 dfrdg4 33416 limsucncmpi 33797 |
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