![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > ordeleqon | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
ordeleqon | ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onprc 7309 | . . . 4 ⊢ ¬ On ∈ V | |
2 | elex 3427 | . . . 4 ⊢ (On ∈ 𝐴 → On ∈ V) | |
3 | 1, 2 | mto 189 | . . 3 ⊢ ¬ On ∈ 𝐴 |
4 | ordon 7308 | . . . . . 6 ⊢ Ord On | |
5 | ordtri3or 6055 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴)) | |
6 | 4, 5 | mpan2 678 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴)) |
7 | df-3or 1069 | . . . . 5 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴) ↔ ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴)) | |
8 | 6, 7 | sylib 210 | . . . 4 ⊢ (Ord 𝐴 → ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴)) |
9 | 8 | ord 850 | . . 3 ⊢ (Ord 𝐴 → (¬ (𝐴 ∈ On ∨ 𝐴 = On) → On ∈ 𝐴)) |
10 | 3, 9 | mt3i 144 | . 2 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On)) |
11 | eloni 6033 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
12 | ordeq 6030 | . . . 4 ⊢ (𝐴 = On → (Ord 𝐴 ↔ Ord On)) | |
13 | 4, 12 | mpbiri 250 | . . 3 ⊢ (𝐴 = On → Ord 𝐴) |
14 | 11, 13 | jaoi 843 | . 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → Ord 𝐴) |
15 | 10, 14 | impbii 201 | 1 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 198 ∨ wo 833 ∨ w3o 1067 = wceq 1507 ∈ wcel 2050 Vcvv 3409 Ord word 6022 Oncon0 6023 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pr 5180 ax-un 7273 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-pss 3839 df-nul 4173 df-if 4345 df-sn 4436 df-pr 4438 df-tp 4440 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-tr 5025 df-eprel 5311 df-po 5320 df-so 5321 df-fr 5360 df-we 5362 df-ord 6026 df-on 6027 |
This theorem is referenced by: ordsson 7314 ssonprc 7317 ordunisuc 7357 orduninsuc 7368 limomss 7395 omon 7401 limom 7405 tfrlem14 7825 tfr2b 7830 unialeph 9315 ordtoplem 33303 ordcmp 33315 |
Copyright terms: Public domain | W3C validator |