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Theorem ordelord 4312
 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 𝐴𝐵𝐴) → Ord 𝐵)

Proof of Theorem ordelord
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 eleq1 2203 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
21anbi2d 460 . . . 4 (𝑥 = 𝐵 → ((Ord 𝐴𝑥𝐴) ↔ (Ord 𝐴𝐵𝐴)))
3 ordeq 4303 . . . 4 (𝑥 = 𝐵 → (Ord 𝑥 ↔ Ord 𝐵))
42, 3imbi12d 233 . . 3 (𝑥 = 𝐵 → (((Ord 𝐴𝑥𝐴) → Ord 𝑥) ↔ ((Ord 𝐴𝐵𝐴) → Ord 𝐵)))
5 dford3 4298 . . . . . 6 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
65simprbi 273 . . . . 5 (Ord 𝐴 → ∀𝑥𝐴 Tr 𝑥)
76r19.21bi 2524 . . . 4 ((Ord 𝐴𝑥𝐴) → Tr 𝑥)
8 ordelss 4310 . . . 4 ((Ord 𝐴𝑥𝐴) → 𝑥𝐴)
9 simpl 108 . . . 4 ((Ord 𝐴𝑥𝐴) → Ord 𝐴)
10 trssord 4311 . . . 4 ((Tr 𝑥𝑥𝐴 ∧ Ord 𝐴) → Ord 𝑥)
117, 8, 9, 10syl3anc 1217 . . 3 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
124, 11vtoclg 2750 . 2 (𝐵𝐴 → ((Ord 𝐴𝐵𝐴) → Ord 𝐵))
1312anabsi7 571 1 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  ∀wral 2417   ⊆ wss 3077  Tr wtr 4035  Ord word 4293 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2692  df-in 3083  df-ss 3090  df-uni 3746  df-tr 4036  df-iord 4297 This theorem is referenced by:  tron  4313  ordelon  4314  ordsucg  4427  ordwe  4499  smores  6198
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