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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 5701 | . . . 4 ⊢ (Lim suc 𝐴 → Ord suc 𝐴) | |
2 | ordsuc 6906 | . . . 4 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
3 | 1, 2 | sylibr 223 | . . 3 ⊢ (Lim suc 𝐴 → Ord 𝐴) |
4 | limuni 5702 | . . 3 ⊢ (Lim suc 𝐴 → suc 𝐴 = ∪ suc 𝐴) | |
5 | ordunisuc 6924 | . . . . 5 ⊢ (Ord 𝐴 → ∪ suc 𝐴 = 𝐴) | |
6 | 5 | eqeq2d 2620 | . . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 ↔ suc 𝐴 = 𝐴)) |
7 | ordirr 5658 | . . . . . 6 ⊢ (Ord 𝐴 → ¬ 𝐴 ∈ 𝐴) | |
8 | eleq2 2677 | . . . . . . 7 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∈ suc 𝐴 ↔ 𝐴 ∈ 𝐴)) | |
9 | 8 | notbid 307 | . . . . . 6 ⊢ (suc 𝐴 = 𝐴 → (¬ 𝐴 ∈ suc 𝐴 ↔ ¬ 𝐴 ∈ 𝐴)) |
10 | 7, 9 | syl5ibrcom 236 | . . . . 5 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ suc 𝐴)) |
11 | sucidg 5720 | . . . . . 6 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ suc 𝐴) | |
12 | 11 | con3i 149 | . . . . 5 ⊢ (¬ 𝐴 ∈ suc 𝐴 → ¬ 𝐴 ∈ 𝑉) |
13 | 10, 12 | syl6 34 | . . . 4 ⊢ (Ord 𝐴 → (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ 𝑉)) |
14 | 6, 13 | sylbid 229 | . . 3 ⊢ (Ord 𝐴 → (suc 𝐴 = ∪ suc 𝐴 → ¬ 𝐴 ∈ 𝑉)) |
15 | 3, 4, 14 | sylc 63 | . 2 ⊢ (Lim suc 𝐴 → ¬ 𝐴 ∈ 𝑉) |
16 | 15 | con2i 133 | 1 ⊢ (𝐴 ∈ 𝑉 → ¬ Lim suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1475 ∈ wcel 1977 ∪ cuni 4372 Ord word 5639 Lim wlim 5641 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 |
This theorem is referenced by: tz7.44-2 7390 rankxpsuc 8628 dfrdg2 30945 dfrdg4 31228 |
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