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Mirrors > Home > MPE Home > Th. List > nlimsucg | Structured version Visualization version GIF version |
Description: A successor is not a limit ordinal. (Contributed by NM, 25-Mar-1995.) (Proof shortened by Andrew Salmon, 27-Aug-2011.) |
Ref | Expression |
---|---|
nlimsucg | ⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limord 6253 | . . . 4 ⊢ (Lim suc 𝐴 → Ord suc 𝐴) | |
2 | ordsuc 7532 | . . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
3 | 1, 2 | sylibr 236 | . . 3 ⊢ (Lim suc 𝐴 → Ord 𝐴) |
4 | limuni 6254 | . . 3 ⊢ (Lim suc 𝐴 → suc 𝐴 = ∪ suc 𝐴) | |
5 | ordunisuc 7550 | . . . . 5 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
6 | 5 | eqeq2d 2835 | . . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 ↔ suc 𝐴 = 𝐴)) |
7 | ordirr 6212 | . . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
8 | eleq2 2904 | . . . . . . 7 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐴)) | |
9 | 8 | notbid 320 | . . . . . 6 ⊢ (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴 ∈ 𝐴)) |
10 | 7, 9 | syl5ibrcom 249 | . . . . 5 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴)) |
11 | sucidg 6272 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) | |
12 | 11 | con3i 157 | . . . . 5 ⊢ (¬ 𝐴 ∈ suc 𝐴 → ¬ 𝐴 ∈ 𝑉) |
13 | 10, 12 | syl6 35 | . . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ 𝑉)) |
14 | 6, 13 | sylbid 242 | . . 3 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 → ¬ 𝐴 ∈ 𝑉)) |
15 | 3, 4, 14 | sylc 65 | . 2 ⊢ (Lim suc 𝐴 → ¬ 𝐴 ∈ 𝑉) |
16 | 15 | con2i 141 | 1 ⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2113 ∪ cuni 4841 Ord word 6193 Lim wlim 6195 suc csuc 6196 |
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 ax-un 7464 |
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-sn 4571 df-pr 4573 df-tp 4575 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-on 6198 df-lim 6199 df-suc 6200 |
This theorem is referenced by: tz7.44-2 8046 rankxpsuc 9314 dfrdg2 33044 dfrdg4 33416 dfsucon 39895 |
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