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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 7789 | . . 3 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) | |
2 | suceq 6388 | . . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵) | |
3 | 2 | eleq1d 2823 | . . . . 5 ⊢ (𝑥 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴)) |
4 | 3 | rspccv 3581 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
5 | 4 | 3ad2ant3 1136 | . . 3 ⊢ ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
6 | 1, 5 | sylbi 216 | . 2 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
7 | limord 6382 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
8 | ordtr 6336 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
9 | trsuc 6409 | . . . 4 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) | |
10 | 9 | ex 414 | . . 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 211 | 1 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ w3a 1088 = wceq 1542 ∈ wcel 2107 ∀wral 3065 ∅c0 4287 Tr wtr 5227 Ord word 6321 Lim wlim 6323 suc csuc 6324 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 ax-un 7677 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-pss 3934 df-nul 4288 df-if 4492 df-pw 4567 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-tr 5228 df-eprel 5542 df-po 5550 df-so 5551 df-fr 5593 df-we 5595 df-ord 6325 df-on 6326 df-lim 6327 df-suc 6328 |
This theorem is referenced by: limsssuc 7791 limuni3 7793 peano2b 7824 rdgsucg 8374 rdgsucmptnf 8380 oesuclem 8476 oaordi 8498 omordi 8518 oeordi 8539 oelim2 8547 limenpsi 9103 r1tr 9719 r1ordg 9721 r1pwss 9727 r1val1 9729 rankdmr1 9744 rankr1bg 9746 pwwf 9750 rankr1c 9764 rankonidlem 9771 ranklim 9787 r1pwcl 9790 rankxplim3 9824 infxpenlem 9956 alephordi 10017 cflm 10193 cfslb2n 10211 alephreg 10525 r1limwun 10679 rankcf 10720 inatsk 10721 oldlim 27238 succlg 41692 |
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