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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 6450 | . . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵) | |
| 3 | 2 | eleq1d 2826 | . . . . 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 1136 | . . 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 6444 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
| 8 | ordtr 6398 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
| 9 | trsuc 6471 | . . . 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 1087 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∅c0 4333 Tr wtr 5259 Ord word 6383 Lim wlim 6385 suc csuc 6386 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 |
| This theorem is referenced by: limsssuc 7871 limuni3 7873 peano2b 7904 rdgsucg 8463 rdgsucmptnf 8469 oesuclem 8563 oaordi 8584 omordi 8604 oeordi 8625 oelim2 8633 limenpsi 9192 r1tr 9816 r1ordg 9818 r1pwss 9824 r1val1 9826 rankdmr1 9841 rankr1bg 9843 pwwf 9847 rankr1c 9861 rankonidlem 9868 ranklim 9884 r1pwcl 9887 rankxplim3 9921 infxpenlem 10053 alephordi 10114 cflm 10290 cfslb2n 10308 alephreg 10622 r1limwun 10776 rankcf 10817 inatsk 10818 oldlim 27925 succlg 43341 |
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