Theorem ordeleqon 7555

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 7551	. . . 4 ⊢ ¬ On ∈ V
2		elex 3419	. . . 4 ⊢ (On ∈ 𝐴 → On ∈ V)
3	1, 2	mto 200	. . 3 ⊢ ¬ On ∈ 𝐴
4		ordon 7550	. . . . . 6 ⊢ Ord On
5		ordtri3or 6234	. . . . . 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 1090	. . . . 5 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴) ↔ ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
8	6, 7	sylib 221	. . . 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 151	. 2 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
11		eloni 6212	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
12		ordeq 6209	. . . 4 ⊢ (𝐴 = On → (Ord 𝐴 ↔ Ord On))
13	4, 12	mpbiri 261	. . 3 ⊢ (𝐴 = On → Ord 𝐴)
14	11, 13	jaoi 857	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → Ord 𝐴)
15	10, 14	impbii 212	1 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))

Syntax hints:   ↔ wb 209   ∨ wo 847   ∨ w3o 1088   = wceq 1543   ∈ wcel 2110  Vcvv 3401  Ord word 6201  Oncon0 6202

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-11 2158  ax-ext 2706  ax-sep 5181  ax-nul 5188  ax-pr 5311  ax-un 7512

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3403  df-dif 3860  df-un 3862  df-in 3864  df-ss 3874  df-pss 3876  df-nul 4228  df-if 4430  df-sn 4532  df-pr 4534  df-tp 4536  df-op 4538  df-uni 4810  df-br 5044  df-opab 5106  df-tr 5151  df-eprel 5449  df-po 5457  df-so 5458  df-fr 5498  df-we 5500  df-ord 6205  df-on 6206

This theorem is referenced by:  ordsson  7556  ssonprc  7560  ordunisuc  7600  orduninsuc  7611  limomss  7638  omon  7645  limom  7649  tfrlem14  8116  tfr2b  8121  unialeph  9698  ordtoplem  34318  ordcmp  34330