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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 6906 | . . 3 ⊢ (Ord 𝐴 ↔ Ord suc 𝐴) | |
2 | sucexb 6901 | . . 3 ⊢ (𝐴 ∈ V ↔ suc 𝐴 ∈ V) | |
3 | 1, 2 | anbi12i 729 | . 2 ⊢ ((Ord 𝐴 ∧ 𝐴 ∈ V) ↔ (Ord suc 𝐴 ∧ suc 𝐴 ∈ V)) |
4 | elon2 5651 | . 2 ⊢ (𝐴 ∈ On ↔ (Ord 𝐴 ∧ 𝐴 ∈ V)) | |
5 | elon2 5651 | . 2 ⊢ (suc 𝐴 ∈ On ↔ (Ord suc 𝐴 ∧ suc 𝐴 ∈ V)) | |
6 | 3, 4, 5 | 3bitr4i 291 | 1 ⊢ (𝐴 ∈ On ↔ suc 𝐴 ∈ On) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∧ wa 383 ∈ wcel 1977 Vcvv 3173 Ord word 5639 Oncon0 5640 suc csuc 5642 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 df-suc 5646 |
This theorem is referenced by: onsucmin 6913 tfindsg2 6953 oaordi 7513 oalimcl 7527 omlimcl 7545 omeulem1 7549 oeordsuc 7561 infensuc 8023 cantnflem1b 8466 cantnflem1 8469 r1ordg 8524 alephnbtwn 8777 cfsuc 8962 alephsuc3 9281 alephreg 9283 nobndlem1 31091 nobndlem8 31098 nofulllem4 31104 nofulllem5 31105 |
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