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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 7080 | . 2 ⊢ Lim ω | |
2 | limsuc 7049 | . 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 196 ∈ wcel 1990 Lim wlim 5724 suc csuc 5725 ωcom 7065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-om 7066 |
This theorem is referenced by: nnsuc 7082 peano2 7086 peano5 7089 frsuc 7532 frsucmptn 7534 nnaordi 7698 nnmsucr 7705 omsmolem 7733 php 8144 php4 8147 unblem1 8212 isfinite2 8218 inf0 8518 inf3lem1 8525 inf3lem5 8529 cantnfp1lem3 8577 cantnflem1 8586 itunisuc 9241 ituniiun 9244 indpi 9729 rdgeqoa 33218 |
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