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Theorem ordelord 4399
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 2252 . . . . 5 (𝑥 = 𝐵 → (𝑥𝐴𝐵𝐴))
21anbi2d 464 . . . 4 (𝑥 = 𝐵 → ((Ord 𝐴𝑥𝐴) ↔ (Ord 𝐴𝐵𝐴)))
3 ordeq 4390 . . . 4 (𝑥 = 𝐵 → (Ord 𝑥 ↔ Ord 𝐵))
42, 3imbi12d 234 . . 3 (𝑥 = 𝐵 → (((Ord 𝐴𝑥𝐴) → Ord 𝑥) ↔ ((Ord 𝐴𝐵𝐴) → Ord 𝐵)))
5 dford3 4385 . . . . . 6 (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥𝐴 Tr 𝑥))
65simprbi 275 . . . . 5 (Ord 𝐴 → ∀𝑥𝐴 Tr 𝑥)
76r19.21bi 2578 . . . 4 ((Ord 𝐴𝑥𝐴) → Tr 𝑥)
8 ordelss 4397 . . . 4 ((Ord 𝐴𝑥𝐴) → 𝑥𝐴)
9 simpl 109 . . . 4 ((Ord 𝐴𝑥𝐴) → Ord 𝐴)
10 trssord 4398 . . . 4 ((Tr 𝑥𝑥𝐴 ∧ Ord 𝐴) → Ord 𝑥)
117, 8, 9, 10syl3anc 1249 . . 3 ((Ord 𝐴𝑥𝐴) → Ord 𝑥)
124, 11vtoclg 2812 . 2 (𝐵𝐴 → ((Ord 𝐴𝐵𝐴) → Ord 𝐵))
1312anabsi7 581 1 ((Ord 𝐴𝐵𝐴) → Ord 𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  wral 2468  wss 3144  Tr wtr 4116  Ord word 4380
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-in 3150  df-ss 3157  df-uni 3825  df-tr 4117  df-iord 4384
This theorem is referenced by:  tron  4400  ordelon  4401  ordsucg  4519  ordwe  4593  smores  6316
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