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Theorem ordelord 4143
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 2114 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
21anbi2d 445 . . . 4 (𝑥 = 𝐵 → ((Ord 𝐴𝑥𝐴) ↔ (Ord 𝐴𝐵𝐴)))
3 ordeq 4134 . . . 4 (𝑥 = 𝐵 → (Ord 𝑥 ↔ Ord 𝐵))
42, 3imbi12d 227 . . 3 (𝑥 = 𝐵 → (((Ord 𝐴𝑥𝐴) → Ord 𝑥) ↔ ((Ord 𝐴𝐵𝐴) → Ord 𝐵)))
5 dford3 4129 . . . . . 6 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
65simprbi 264 . . . . 5 (Ord 𝐴 → ∀𝑥𝐴 Tr 𝑥)
76r19.21bi 2422 . . . 4 ((Ord 𝐴𝑥𝐴) → Tr 𝑥)
8 ordelss 4141 . . . 4 ((Ord 𝐴𝑥𝐴) → 𝑥𝐴)
9 simpl 106 . . . 4 ((Ord 𝐴𝑥𝐴) → Ord 𝐴)
10 trssord 4142 . . . 4 ((Tr 𝑥𝑥𝐴 ∧ Ord 𝐴) → Ord 𝑥)
117, 8, 9, 10syl3anc 1144 . . 3 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
124, 11vtoclg 2628 . 2 (𝐵𝐴 → ((Ord 𝐴𝐵𝐴) → Ord 𝐵))
1312anabsi7 523 1 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1257  wcel 1407  wral 2321  wss 2942  Tr wtr 3879  Ord word 4124
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-v 2574  df-in 2949  df-ss 2956  df-uni 3606  df-tr 3880  df-iord 4128
This theorem is referenced by:  tron  4144  ordelon  4145  ordsucg  4253  ordwe  4325  smores  5935
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