Theorem ordeleqon 7800

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 7796	. . . 4 ⊢ ¬ On ∈ V
2		elex 3498	. . . 4 ⊢ (On ∈ 𝐴 → On ∈ V)
3	1, 2	mto 197	. . 3 ⊢ ¬ On ∈ 𝐴
4		ordon 7795	. . . . . 6 ⊢ Ord On
5		ordtri3or 6417	. . . . . 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 6395	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
12		ordeq 6392	. . . 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 1536   ∈ wcel 2105  Vcvv 3477  Ord word 6384  Oncon0 6385

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  df-ord 6388  df-on 6389

This theorem is referenced by:  ordsson  7801  ssonprc  7806  ordunisuc  7851  orduninsuc  7863  limomss  7891  omon  7898  limom  7902  tfrlem14  8429  tfr2b  8434  unialeph  10138  ordtoplem  36417  ordcmp  36429  onsupnmax  43216  dflim5  43318