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Mirrors > Home > MPE Home > Th. List > limensuc | Structured version Visualization version GIF version |
Description: A limit ordinal is equinumerous to its successor. (Contributed by NM, 30-Oct-2003.) |
Ref | Expression |
---|---|
limensuc | ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ≈ suc 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2689 | . . . 4 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → (𝐴 ∈ 𝑉 ↔ if(Lim 𝐴, 𝐴, On) ∈ 𝑉)) | |
2 | id 22 | . . . . 5 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → 𝐴 = if(Lim 𝐴, 𝐴, On)) | |
3 | suceq 5790 | . . . . 5 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → suc 𝐴 = suc if(Lim 𝐴, 𝐴, On)) | |
4 | 2, 3 | breq12d 4666 | . . . 4 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → (𝐴 ≈ suc 𝐴 ↔ if(Lim 𝐴, 𝐴, On) ≈ suc if(Lim 𝐴, 𝐴, On))) |
5 | 1, 4 | imbi12d 334 | . . 3 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → ((𝐴 ∈ 𝑉 → 𝐴 ≈ suc 𝐴) ↔ (if(Lim 𝐴, 𝐴, On) ∈ 𝑉 → if(Lim 𝐴, 𝐴, On) ≈ suc if(Lim 𝐴, 𝐴, On)))) |
6 | limeq 5735 | . . . . 5 ⊢ (𝐴 = if(Lim 𝐴, 𝐴, On) → (Lim 𝐴 ↔ Lim if(Lim 𝐴, 𝐴, On))) | |
7 | limeq 5735 | . . . . 5 ⊢ (On = if(Lim 𝐴, 𝐴, On) → (Lim On ↔ Lim if(Lim 𝐴, 𝐴, On))) | |
8 | limon 7036 | . . . . 5 ⊢ Lim On | |
9 | 6, 7, 8 | elimhyp 4146 | . . . 4 ⊢ Lim if(Lim 𝐴, 𝐴, On) |
10 | 9 | limensuci 8136 | . . 3 ⊢ (if(Lim 𝐴, 𝐴, On) ∈ 𝑉 → if(Lim 𝐴, 𝐴, On) ≈ suc if(Lim 𝐴, 𝐴, On)) |
11 | 5, 10 | dedth 4139 | . 2 ⊢ (Lim 𝐴 → (𝐴 ∈ 𝑉 → 𝐴 ≈ suc 𝐴)) |
12 | 11 | impcom 446 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) → 𝐴 ≈ suc 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∈ wcel 1990 ifcif 4086 class class class wbr 4653 Oncon0 5723 Lim wlim 5724 suc csuc 5725 ≈ cen 7952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-1o 7560 df-er 7742 df-en 7956 df-dom 7957 |
This theorem is referenced by: infensuc 8138 |
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