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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 4338 | . . 3 ⊢ ¬ ∅ ∈ ∅ | |
| 2 | simp2 1138 | . . 3 ⊢ ((Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅) → ∅ ∈ ∅) | |
| 3 | 1, 2 | mto 197 | . 2 ⊢ ¬ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅) |
| 4 | dflim2 6441 | . 2 ⊢ (Lim ∅ ↔ (Ord ∅ ∧ ∅ ∈ ∅ ∧ ∅ = ∪ ∅)) | |
| 5 | 3, 4 | mtbir 323 | 1 ⊢ ¬ Lim ∅ |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 ∧ w3a 1087 = wceq 1540 ∈ wcel 2108 ∅c0 4333 ∪ cuni 4907 Ord word 6383 Lim wlim 6385 |
| 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 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-lim 6389 |
| This theorem is referenced by: 0ellim 6447 tz7.44lem1 8445 tz7.44-3 8448 1ellim 8536 2ellim 8537 cflim2 10303 rankcf 10817 dfrdg4 35952 limsucncmpi 36446 onov0suclim 43287 |
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