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Mirrors > Home > MPE Home > Th. List > sucelon | Structured version Visualization version GIF version |
Description: The successor of an ordinal number is an ordinal number. (Contributed by NM, 9-Sep-2003.) |
Ref | Expression |
---|---|
sucelon | ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsuc 7518 | . . 3 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
2 | sucexb 7513 | . . 3 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
3 | 1, 2 | anbi12i 626 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) ↔ (Ord suc 𝐴 ∧ suc 𝐴 ∈ V)) |
4 | elon2 6195 | . 2 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) | |
5 | elon2 6195 | . 2 ⊢ (suc 𝐴 ∈ On ↔ (Ord suc 𝐴 ∧ suc 𝐴 ∈ V)) | |
6 | 3, 4, 5 | 3bitr4i 304 | 1 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∧ wa 396 ∈ wcel 2105 Vcvv 3492 Ord word 6183 Oncon0 6184 suc csuc 6186 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-suc 6190 |
This theorem is referenced by: onsucmin 7525 tfindsg2 7565 oaordi 8161 oalimcl 8175 omlimcl 8193 omeulem1 8197 oeordsuc 8209 infensuc 8683 cantnflem1b 9137 cantnflem1 9140 r1ordg 9195 alephnbtwn 9485 cfsuc 9667 alephsuc3 9990 alephreg 9992 bdayimaon 33094 nosupbnd1lem1 33105 nosupbnd1 33111 nosupbnd2lem1 33112 nosupbnd2 33113 |
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