Theorem ordeleqon 7758

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. (Contributed by NM, 1-Jun-2003.)

Assertion

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

Proof of Theorem ordeleqon

Step	Hyp	Ref	Expression
1		onprc 7754	. . . 4 ⊢ ¬ On ∈ V
2		elex 3468	. . . 4 ⊢ (On ∈ 𝐴 → On ∈ V)
3	1, 2	mto 197	. . 3 ⊢ ¬ On ∈ 𝐴
4		ordon 7753	. . . . . 6 ⊢ Ord On
5		ordtri3or 6364	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴))
6	4, 5	mpan2 691	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴))
7		df-3or 1087	. . . . 5 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴) ↔ ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
8	6, 7	sylib 218	. . . 4 ⊢ (Ord 𝐴 → ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
9	8	ord 864	. . 3 ⊢ (Ord 𝐴 → (¬ (𝐴 ∈ On ∨ 𝐴 = On) → On ∈ 𝐴))
10	3, 9	mt3i 149	. 2 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
11		eloni 6342	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
12		ordeq 6339	. . . 4 ⊢ (𝐴 = On → (Ord 𝐴 ↔ Ord On))
13	4, 12	mpbiri 258	. . 3 ⊢ (𝐴 = On → Ord 𝐴)
14	11, 13	jaoi 857	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → Ord 𝐴)
15	10, 14	impbii 209	1 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))

Syntax hints:   ↔ wb 206   ∨ wo 847   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109  Vcvv 3447  Ord word 6331  Oncon0 6332

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3934  df-nul 4297  df-if 4489  df-pw 4565  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-tr 5215  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5591  df-we 5593  df-ord 6335  df-on 6336

This theorem is referenced by:  ordsson  7759  ssonprc  7763  ordunisuc  7807  orduninsuc  7819  limomss  7847  omon  7854  limom  7858  tfrlem14  8359  tfr2b  8364  unialeph  10054  ordtoplem  36423  ordcmp  36435  onsupnmax  43217  dflim5  43318