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Mirrors > Home > ILE Home > Th. List > ordsucg | GIF version |
Description: The successor of an ordinal class is ordinal. (Contributed by Jim Kingdon, 20-Nov-2018.) |
Ref | Expression |
---|---|
ordsucg | ⊢ (A ∈ V → (Ord A ↔ Ord suc A)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsucim 4192 | . 2 ⊢ (Ord A → Ord suc A) | |
2 | sucidg 4119 | . . 3 ⊢ (A ∈ V → A ∈ suc A) | |
3 | ordelord 4084 | . . . 4 ⊢ ((Ord suc A ∧ A ∈ suc A) → Ord A) | |
4 | 3 | ex 108 | . . 3 ⊢ (Ord suc A → (A ∈ suc A → Ord A)) |
5 | 2, 4 | syl5com 26 | . 2 ⊢ (A ∈ V → (Ord suc A → Ord A)) |
6 | 1, 5 | impbid2 131 | 1 ⊢ (A ∈ V → (Ord A ↔ Ord suc A)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 98 ∈ wcel 1390 Vcvv 2551 Ord word 4065 suc csuc 4068 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-un 2916 df-in 2918 df-ss 2925 df-sn 3373 df-uni 3572 df-tr 3846 df-iord 4069 df-suc 4074 |
This theorem is referenced by: sucelon 4195 |
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