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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 7869 | . . 3 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) | |
2 | suceq 6452 | . . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵) | |
3 | 2 | eleq1d 2824 | . . . . 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 1134 | . . 3 ⊢ ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
6 | 1, 5 | sylbi 217 | . 2 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
7 | limord 6446 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
8 | ordtr 6400 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
9 | trsuc 6473 | . . . 4 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) | |
10 | 9 | ex 412 | . . 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 212 | 1 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∅c0 4339 Tr wtr 5265 Ord word 6385 Lim wlim 6387 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 |
This theorem is referenced by: limsssuc 7871 limuni3 7873 peano2b 7904 rdgsucg 8462 rdgsucmptnf 8468 oesuclem 8562 oaordi 8583 omordi 8603 oeordi 8624 oelim2 8632 limenpsi 9191 r1tr 9814 r1ordg 9816 r1pwss 9822 r1val1 9824 rankdmr1 9839 rankr1bg 9841 pwwf 9845 rankr1c 9859 rankonidlem 9866 ranklim 9882 r1pwcl 9885 rankxplim3 9919 infxpenlem 10051 alephordi 10112 cflm 10288 cfslb2n 10306 alephreg 10620 r1limwun 10774 rankcf 10815 inatsk 10816 oldlim 27940 succlg 43318 |
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