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Mirrors > Home > MPE Home > Th. List > limsuc | Structured version Visualization version GIF version |
Description: The successor of a member of a limit ordinal is also a member. (Contributed by NM, 3-Sep-2003.) |
Ref | Expression |
---|---|
limsuc | ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dflim4 7551 | . . 3 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) | |
2 | suceq 6250 | . . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵) | |
3 | 2 | eleq1d 2897 | . . . . 5 ⊢ (𝑥 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴)) |
4 | 3 | rspccv 3619 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
5 | 4 | 3ad2ant3 1127 | . . 3 ⊢ ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
6 | 1, 5 | sylbi 218 | . 2 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
7 | limord 6244 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
8 | ordtr 6199 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
9 | trsuc 6269 | . . . 4 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) | |
10 | 9 | ex 413 | . . 3 ⊢ (Tr 𝐴 → (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
11 | 7, 8, 10 | 3syl 18 | . 2 ⊢ (Lim 𝐴 → (suc 𝐵 ∈ 𝐴 → 𝐵 ∈ 𝐴)) |
12 | 6, 11 | impbid 213 | 1 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 207 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3138 ∅c0 4290 Tr wtr 5164 Ord word 6184 Lim wlim 6186 suc csuc 6187 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-tr 5165 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 |
This theorem is referenced by: limsssuc 7553 limuni3 7555 peano2b 7584 rdgsucg 8050 rdgsucmptnf 8056 oesuclem 8141 oaordi 8162 omordi 8182 oeordi 8203 oelim2 8211 limenpsi 8681 r1tr 9194 r1ordg 9196 r1pwss 9202 r1val1 9204 rankdmr1 9219 rankr1bg 9221 pwwf 9225 rankr1c 9239 rankonidlem 9246 ranklim 9262 r1pwcl 9265 rankxplim3 9299 infxpenlem 9428 alephordi 9489 cflm 9661 cfslb2n 9679 alephreg 9993 r1limwun 10147 rankcf 10188 inatsk 10189 |
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