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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 7454 | . 2 ⊢ Lim ω | |
2 | limsuc 7423 | . 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 207 ∈ wcel 2080 Lim wlim 6070 suc csuc 6071 ωcom 7439 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pr 5224 ax-un 7322 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-ral 3109 df-rex 3110 df-rab 3113 df-v 3438 df-sbc 3708 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-br 4965 df-opab 5027 df-tr 5067 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-om 7440 |
This theorem is referenced by: nnsuc 7456 peano2 7461 peano5 7464 frsuc 7927 frsucmptn 7929 nnaordi 8097 nnmsucr 8104 omsmolem 8133 php 8551 php4 8554 unblem1 8619 isfinite2 8625 inf0 8933 inf3lem1 8940 inf3lem5 8944 cantnfp1lem3 8992 cantnflem1 9001 itunisuc 9690 ituniiun 9693 indpi 10178 rdgeqoa 34195 |
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