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Mirrors > Home > MPE Home > Th. List > nlimon | Structured version Visualization version GIF version |
Description: Two ways to express the class of non-limit ordinal numbers. Part of Definition 7.27 of [TakeutiZaring] p. 42, who use the symbol KI for this class. (Contributed by NM, 1-Nov-2004.) |
Ref | Expression |
---|---|
nlimon | ⊢ {𝑥 ∈ On ∣ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)} = {𝑥 ∈ On ∣ ¬ Lim 𝑥} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eloni 6396 | . . 3 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
2 | dflim3 7868 | . . . . 5 ⊢ (Lim 𝑥 ↔ (Ord 𝑥 ∧ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦))) | |
3 | 2 | baib 535 | . . . 4 ⊢ (Ord 𝑥 → (Lim 𝑥 ↔ ¬ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦))) |
4 | 3 | con2bid 354 | . . 3 ⊢ (Ord 𝑥 → ((𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦) ↔ ¬ Lim 𝑥)) |
5 | 1, 4 | syl 17 | . 2 ⊢ (𝑥 ∈ On → ((𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦) ↔ ¬ Lim 𝑥)) |
6 | 5 | rabbiia 3437 | 1 ⊢ {𝑥 ∈ On ∣ (𝑥 = ∅ ∨ ∃𝑦 ∈ On 𝑥 = suc 𝑦)} = {𝑥 ∈ On ∣ ¬ Lim 𝑥} |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 206 ∨ wo 847 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 {crab 3433 ∅c0 4339 Ord word 6385 Oncon0 6386 Lim wlim 6387 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 |
This theorem is referenced by: (None) |
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