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Theorem ordelord 4084
Description: An element of an ordinal class is ordinal. Proposition 7.6 of [TakeutiZaring] p. 36. (Contributed by NM, 23-Apr-1994.)
Assertion
Ref Expression
ordelord ((Ord A B A) → Ord B)

Proof of Theorem ordelord
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2097 . . . . 5 (x = B → (x AB A))
21anbi2d 437 . . . 4 (x = B → ((Ord A x A) ↔ (Ord A B A)))
3 ordeq 4075 . . . 4 (x = B → (Ord x ↔ Ord B))
42, 3imbi12d 223 . . 3 (x = B → (((Ord A x A) → Ord x) ↔ ((Ord A B A) → Ord B)))
5 dford3 4070 . . . . . 6 (Ord A ↔ (Tr A x A Tr x))
65simprbi 260 . . . . 5 (Ord Ax A Tr x)
76r19.21bi 2401 . . . 4 ((Ord A x A) → Tr x)
8 ordelss 4082 . . . 4 ((Ord A x A) → xA)
9 simpl 102 . . . 4 ((Ord A x A) → Ord A)
10 trssord 4083 . . . 4 ((Tr x xA Ord A) → Ord x)
117, 8, 9, 10syl3anc 1134 . . 3 ((Ord A x A) → Ord x)
124, 11vtoclg 2607 . 2 (B A → ((Ord A B A) → Ord B))
1312anabsi7 515 1 ((Ord A B A) → Ord B)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242   wcel 1390  wral 2300  wss 2911  Tr wtr 3845  Ord word 4065
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-in 2918  df-ss 2925  df-uni 3572  df-tr 3846  df-iord 4069
This theorem is referenced by:  tron  4085  ordelon  4086  ordsucg  4194  smores  5848
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