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 6984 | . . . 4 ⊢ ¬ On ∈ V | |
2 | elex 3212 | . . . 4 ⊢ (On ∈ 𝐴 → On ∈ V) | |
3 | 1, 2 | mto 188 | . . 3 ⊢ ¬ On ∈ 𝐴 |
4 | ordon 6982 | . . . . . 6 ⊢ Ord On | |
5 | ordtri3or 5755 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴)) | |
6 | 4, 5 | mpan2 707 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴)) |
7 | df-3or 1038 | . . . . 5 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴) ↔ ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴)) | |
8 | 6, 7 | sylib 208 | . . . 4 ⊢ (Ord 𝐴 → ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴)) |
9 | 8 | ord 392 | . . 3 ⊢ (Ord 𝐴 → (¬ (𝐴 ∈ On ∨ 𝐴 = On) → On ∈ 𝐴)) |
10 | 3, 9 | mt3i 141 | . 2 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On)) |
11 | eloni 5733 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
12 | ordeq 5730 | . . . 4 ⊢ (𝐴 = On → (Ord 𝐴 ↔ Ord On)) | |
13 | 4, 12 | mpbiri 248 | . . 3 ⊢ (𝐴 = On → Ord 𝐴) |
14 | 11, 13 | jaoi 394 | . 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → Ord 𝐴) |
15 | 10, 14 | impbii 199 | 1 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 196 ∨ wo 383 ∨ w3o 1036 = wceq 1483 ∈ wcel 1990 Vcvv 3200 Ord word 5722 Oncon0 5723 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 |
This theorem is referenced by: ordsson 6989 ssonprc 6992 ordunisuc 7032 orduninsuc 7043 limomss 7070 omon 7076 limom 7080 tfrlem14 7487 tfr2b 7492 unialeph 8924 ordtoplem 32434 ordcmp 32446 |
Copyright terms: Public domain | W3C validator |