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Theorem ordwe 4460
Description: Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.)
Assertion
Ref Expression
ordwe (Ord 𝐴 → E We 𝐴)

Proof of Theorem ordwe
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordfr 4459 . 2 (Ord 𝐴 → E Fr 𝐴)
2 ordelord 4273 . . . . 5 ((Ord 𝐴𝑧𝐴) → Ord 𝑧)
323ad2antr3 1133 . . . 4 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → Ord 𝑧)
4 ordtr1 4280 . . . . 5 (Ord 𝑧 → ((𝑥𝑦𝑦𝑧) → 𝑥𝑧))
5 epel 4184 . . . . . 6 (𝑥 E 𝑦𝑥𝑦)
6 epel 4184 . . . . . 6 (𝑦 E 𝑧𝑦𝑧)
75, 6anbi12i 455 . . . . 5 ((𝑥 E 𝑦𝑦 E 𝑧) ↔ (𝑥𝑦𝑦𝑧))
8 epel 4184 . . . . 5 (𝑥 E 𝑧𝑥𝑧)
94, 7, 83imtr4g 204 . . . 4 (Ord 𝑧 → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
103, 9syl 14 . . 3 ((Ord 𝐴 ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
1110ralrimivvva 2492 . 2 (Ord 𝐴 → ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧))
12 df-wetr 4226 . 2 ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ ∀𝑥𝐴𝑦𝐴𝑧𝐴 ((𝑥 E 𝑦𝑦 E 𝑧) → 𝑥 E 𝑧)))
131, 11, 12sylanbrc 413 1 (Ord 𝐴 → E We 𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  w3a 947  wcel 1465  wral 2393   class class class wbr 3899   E cep 4179   Fr wfr 4220   We wwe 4222  Ord word 4254
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 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-pow 4068  ax-pr 4101  ax-setind 4422
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-un 3045  df-in 3047  df-ss 3054  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-tr 3997  df-eprel 4181  df-frfor 4223  df-frind 4224  df-wetr 4226  df-iord 4258
This theorem is referenced by:  nnwetri  6772
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