Theorem ordeleqon 7766

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 7762	. . . 4 ⊢ ¬ On ∈ V
2		elex 3487	. . . 4 ⊢ (On ∈ 𝐴 → On ∈ V)
3	1, 2	mto 196	. . 3 ⊢ ¬ On ∈ 𝐴
4		ordon 7761	. . . . . 6 ⊢ Ord On
5		ordtri3or 6390	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴))
6	4, 5	mpan2 688	. . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴))
7		df-3or 1085	. . . . 5 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴) ↔ ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
8	6, 7	sylib 217	. . . 4 ⊢ (Ord 𝐴 → ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴))
9	8	ord 861	. . 3 ⊢ (Ord 𝐴 → (¬ (𝐴 ∈ On ∨ 𝐴 = On) → On ∈ 𝐴))
10	3, 9	mt3i 149	. 2 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On))
11		eloni 6368	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
12		ordeq 6365	. . . 4 ⊢ (𝐴 = On → (Ord 𝐴 ↔ Ord On))
13	4, 12	mpbiri 258	. . 3 ⊢ (𝐴 = On → Ord 𝐴)
14	11, 13	jaoi 854	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → Ord 𝐴)
15	10, 14	impbii 208	1 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))

Syntax hints:   ↔ wb 205   ∨ wo 844   ∨ w3o 1083   = wceq 1533   ∈ wcel 2098  Vcvv 3468  Ord word 6357  Oncon0 6358

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-tr 5259  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-ord 6361  df-on 6362

This theorem is referenced by:  ordsson  7767  ssonprc  7772  ordunisuc  7817  orduninsuc  7829  limomss  7857  omon  7864  limom  7868  tfrlem14  8392  tfr2b  8397  unialeph  10098  ordtoplem  35828  ordcmp  35840  onsupnmax  42553  dflim5  42655