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Mirrors > Home > MPE Home > Th. List > peano2b | Structured version Visualization version GIF version |
Description: A class belongs to omega iff its successor does. (Contributed by NM, 3-Dec-1995.) |
Ref | Expression |
---|---|
peano2b | ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limom 7660 | . 2 ⊢ Lim ω | |
2 | limsuc 7628 | . 2 ⊢ (Lim ω → (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω)) | |
3 | 1, 2 | ax-mp 5 | 1 ⊢ (𝐴 ∈ ω ↔ suc 𝐴 ∈ ω) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 209 ∈ wcel 2110 Lim wlim 6214 suc csuc 6215 ωcom 7644 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-11 2158 ax-ext 2708 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-sb 2071 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3410 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-pss 3885 df-nul 4238 df-if 4440 df-pw 4515 df-sn 4542 df-pr 4544 df-tp 4546 df-op 4548 df-uni 4820 df-br 5054 df-opab 5116 df-tr 5162 df-eprel 5460 df-po 5468 df-so 5469 df-fr 5509 df-we 5511 df-ord 6216 df-on 6217 df-lim 6218 df-suc 6219 df-om 7645 |
This theorem is referenced by: nnsuc 7662 peano2 7668 peano5 7671 peano5OLD 7672 frsuc 8172 frsucmptn 8174 nnaordi 8346 nnmsucr 8353 omsmolem 8382 php 8830 php4 8833 unblem1 8923 isfinite2 8929 inf0 9236 inf3lem1 9243 inf3lem5 9247 cantnfp1lem3 9295 cantnflem1 9304 itunisuc 10033 ituniiun 10036 indpi 10521 rdgeqoa 35278 |
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