Theorem ordeleqon 2241

Description: A way to express the ordinal property of a class in terms of the class of ordinal numbers. Corollary 7.14 of [TakeutiZaring] p. 38 and its converse.

Assertion

Ref	Expression
ordeleqon	⊢ (Ord A ↔ (A ∈ On ∨ A = On))

Proof of Theorem ordeleqon

Step	Hyp	Ref	Expression
1		onprc 2240	. . . 4 ⊢ ¬ On ∈ V
2		elisset 1354	. . . 4 ⊢ (On ∈ A → On ∈ V)
3	1, 2	mto 93	. . 3 ⊢ ¬ On ∈ A
4		ordon 2238	. . . . . 6 ⊢ Ord On
5		ordtri3or 2230	. . . . . 6 ⊢ ((Ord A ∧ Ord On) → (A ∈ On ∨ A = On ∨ On ∈ A))
6	4, 5	mpan2 519	. . . . 5 ⊢ (Ord A → (A ∈ On ∨ A = On ∨ On ∈ A))
7		df-3or 582	. . . . 5 ⊢ ((A ∈ On ∨ A = On ∨ On ∈ A) ↔ ((A ∈ On ∨ A = On) ∨ On ∈ A))
8	6, 7	sylib 173	. . . 4 ⊢ (Ord A → ((A ∈ On ∨ A = On) ∨ On ∈ A))
9	8	ord 202	. . 3 ⊢ (Ord A → (¬ (A ∈ On ∨ A = On) → On ∈ A))
10	3, 9	mt3i 100	. 2 ⊢ (Ord A → (A ∈ On ∨ A = On))
11		eloni 2209	. . 3 ⊢ (A ∈ On → Ord A)
12		ordeq 2206	. . . 4 ⊢ (A = On → (Ord A ↔ Ord On))
13	4, 12	mpbiri 169	. . 3 ⊢ (A = On → Ord A)
14	11, 13	jaoi 275	. 2 ⊢ ((A ∈ On ∨ A = On) → Ord A)
15	10, 14	impbi 139	1 ⊢ (Ord A ↔ (A ∈ On ∨ A = On))

Syntax hints:   ↔ wb 127   ∨ wo 195   ∨ w3o 580   = wceq 1091   ∈ wcel 1092  Vcvv 1348  Ord word 2198  Oncon0 2199

This theorem is referenced by:  ordsson 2242  ordunisuc 2339  limsuc 2361  orduninsuc 2365  limomss 2378  omon 2384  limom 2387  tfrlem13 2961

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203

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