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Mirrors > Home > MPE Home > Th. List > onsucb | Structured version Visualization version GIF version |
Description: A class is an ordinal number if and only if its successor is an ordinal number. Biconditional form of onsuc 7782. (Contributed by NM, 9-Sep-2003.) |
Ref | Expression |
---|---|
onsucb | ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsuc 7784 | . . 3 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
2 | sucexb 7775 | . . 3 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
3 | 1, 2 | anbi12i 627 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) ↔ (Ord suc 𝐴 ∧ suc 𝐴 ∈ V)) |
4 | elon2 6364 | . 2 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) | |
5 | elon2 6364 | . 2 ⊢ (suc 𝐴 ∈ On ↔ (Ord suc 𝐴 ∧ suc 𝐴 ∈ V)) | |
6 | 3, 4, 5 | 3bitr4i 302 | 1 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∈ wcel 2106 Vcvv 3473 Ord word 6352 Oncon0 6353 suc csuc 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 ax-un 7708 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3963 df-nul 4319 df-if 4523 df-pw 4598 df-sn 4623 df-pr 4625 df-op 4629 df-uni 4902 df-br 5142 df-opab 5204 df-tr 5259 df-eprel 5573 df-po 5581 df-so 5582 df-fr 5624 df-we 5626 df-ord 6356 df-on 6357 df-suc 6359 |
This theorem is referenced by: onsucmin 7792 tfindsg2 7834 oaordi 8529 oalimcl 8543 omlimcl 8561 omeulem1 8565 oeordsuc 8577 naddcllem 8658 infensuc 9138 cantnflem1b 9663 cantnflem1 9666 r1ordg 9755 alephnbtwn 10048 cfsuc 10234 alephsuc3 10557 alephreg 10559 bdayimaon 27123 nosupbnd1lem1 27138 nosupbnd1 27144 nosupbnd2lem1 27145 nosupbnd2 27146 noinfno 27148 noinfres 27152 noinfbnd1lem1 27153 noinfbnd1 27159 noinfbnd2lem1 27160 noinfbnd2 27161 noeta2 27212 etasslt2 27241 |
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