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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 6397 | . 2 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) | |
5 | elon2 6397 | . 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 2106 Vcvv 3478 Ord word 6385 Oncon0 6386 suc csuc 6388 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-ord 6389 df-on 6390 df-suc 6392 |
This theorem is referenced by: onsucmin 7841 tfindsg2 7883 oaordi 8583 oalimcl 8597 omlimcl 8615 omeulem1 8619 oeordsuc 8631 naddcllem 8713 infensuc 9194 cantnflem1b 9724 cantnflem1 9727 r1ordg 9816 alephnbtwn 10109 cfsuc 10295 alephsuc3 10618 alephreg 10620 bdayimaon 27753 nosupbnd1lem1 27768 nosupbnd1 27774 nosupbnd2lem1 27775 nosupbnd2 27776 noinfno 27778 noinfres 27782 noinfbnd1lem1 27783 noinfbnd1 27789 noinfbnd2lem1 27790 noinfbnd2 27791 noeta2 27844 etasslt2 27874 |
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