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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 7563 | . . 3 ⊢ (Lim 𝐴 ↔ (Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴)) | |
2 | suceq 6256 | . . . . . 6 ⊢ (𝑥 = 𝐵 → suc 𝑥 = suc 𝐵) | |
3 | 2 | eleq1d 2897 | . . . . 5 ⊢ (𝑥 = 𝐵 → (suc 𝑥 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴)) |
4 | 3 | rspccv 3620 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
5 | 4 | 3ad2ant3 1131 | . . 3 ⊢ ((Ord 𝐴 ∧ ∅ ∈ 𝐴 ∧ ∀𝑥 ∈ 𝐴 suc 𝑥 ∈ 𝐴) → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
6 | 1, 5 | sylbi 219 | . 2 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 → suc 𝐵 ∈ 𝐴)) |
7 | limord 6250 | . . 3 ⊢ (Lim 𝐴 → Ord 𝐴) | |
8 | ordtr 6205 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
9 | trsuc 6275 | . . . 4 ⊢ ((Tr 𝐴 ∧ suc 𝐵 ∈ 𝐴) → 𝐵 ∈ 𝐴) | |
10 | 9 | ex 415 | . . 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 214 | 1 ⊢ (Lim 𝐴 → (𝐵 ∈ 𝐴 ↔ suc 𝐵 ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ w3a 1083 = wceq 1537 ∈ wcel 2114 ∀wral 3138 ∅c0 4291 Tr wtr 5172 Ord word 6190 Lim wlim 6192 suc csuc 6193 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 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 3496 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-tr 5173 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 |
This theorem is referenced by: limsssuc 7565 limuni3 7567 peano2b 7596 rdgsucg 8059 rdgsucmptnf 8065 oesuclem 8150 oaordi 8172 omordi 8192 oeordi 8213 oelim2 8221 limenpsi 8692 r1tr 9205 r1ordg 9207 r1pwss 9213 r1val1 9215 rankdmr1 9230 rankr1bg 9232 pwwf 9236 rankr1c 9250 rankonidlem 9257 ranklim 9273 r1pwcl 9276 rankxplim3 9310 infxpenlem 9439 alephordi 9500 cflm 9672 cfslb2n 9690 alephreg 10004 r1limwun 10158 rankcf 10199 inatsk 10200 |
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