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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 7903 | . 2 ⊢ Lim ω | |
| 2 | limsuc 7870 | . 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 206 ∈ wcel 2108 Lim wlim 6385 suc csuc 6386 ωcom 7887 |
| 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 df-om 7888 |
| This theorem is referenced by: nnsuc 7905 peano2 7912 peano5 7915 frsuc 8477 frsucmptn 8479 nnaordi 8656 nnmsucr 8663 omsmolem 8695 php 9247 php4 9250 phpOLD 9259 unblem1 9328 isfinite2 9334 inf0 9661 inf3lem1 9668 inf3lem5 9672 cantnfp1lem3 9720 cantnflem1 9729 itunisuc 10459 ituniiun 10462 indpi 10947 rdgeqoa 37371 |
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