Theorem ordeleqon 7715

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 7711	. . . 4 ⊢ ¬ On ∈ V
2		elex 3457	. . . 4 ⊢ (On ∈ 𝐴 → On ∈ V)
3	1, 2	mto 197	. . 3 ⊢ ¬ On ∈ 𝐴
4		ordon 7710	. . . . . 6 ⊢ Ord On
5		ordtri3or 6338	. . . . . 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 6316	. . 3 ⊢ (𝐴 ∈ On → Ord 𝐴)
12		ordeq 6313	. . . 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 1541   ∈ wcel 2111  Vcvv 3436  Ord word 6305  Oncon0 6306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-tr 5197  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-we 5569  df-ord 6309  df-on 6310

This theorem is referenced by:  ordsson  7716  ssonprc  7720  ordunisuc  7762  orduninsuc  7773  limomss  7801  omon  7808  limom  7812  tfrlem14  8310  tfr2b  8315  unialeph  9992  ordtoplem  36479  ordcmp  36491  onsupnmax  43331  dflim5  43432