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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 7831. (Contributed by NM, 9-Sep-2003.) |
| Ref | Expression |
|---|---|
| onsucb | ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ordsuc 7833 | . . 3 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
| 2 | sucexb 7824 | . . 3 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
| 3 | 1, 2 | anbi12i 628 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) ↔ (Ord suc 𝐴 ∧ suc 𝐴 ∈ V)) |
| 4 | elon2 6395 | . 2 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) | |
| 5 | elon2 6395 | . 2 ⊢ (suc 𝐴 ∈ On ↔ (Ord suc 𝐴 ∧ suc 𝐴 ∈ V)) | |
| 6 | 3, 4, 5 | 3bitr4i 303 | 1 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 ∈ wcel 2108 Vcvv 3480 Ord word 6383 Oncon0 6384 suc csuc 6386 |
| 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-suc 6390 |
| This theorem is referenced by: onsucmin 7841 tfindsg2 7883 oaordi 8584 oalimcl 8598 omlimcl 8616 omeulem1 8620 oeordsuc 8632 naddcllem 8714 infensuc 9195 cantnflem1b 9726 cantnflem1 9729 r1ordg 9818 alephnbtwn 10111 cfsuc 10297 alephsuc3 10620 alephreg 10622 bdayimaon 27738 nosupbnd1lem1 27753 nosupbnd1 27759 nosupbnd2lem1 27760 nosupbnd2 27761 noinfno 27763 noinfres 27767 noinfbnd1lem1 27768 noinfbnd1 27774 noinfbnd2lem1 27775 noinfbnd2 27776 noeta2 27829 etasslt2 27859 |
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