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| 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 7798 | . . . 4 ⊢ ¬ On ∈ V | |
| 2 | elex 3501 | . . . 4 ⊢ (On ∈ 𝐴 → On ∈ V) | |
| 3 | 1, 2 | mto 197 | . . 3 ⊢ ¬ On ∈ 𝐴 |
| 4 | ordon 7797 | . . . . . 6 ⊢ Ord On | |
| 5 | ordtri3or 6416 | . . . . . 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 1088 | . . . . 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 865 | . . 3 ⊢ (Ord 𝐴 → (¬ (𝐴 ∈ On ∨ 𝐴 = On) → On ∈ 𝐴)) |
| 10 | 3, 9 | mt3i 149 | . 2 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On)) |
| 11 | eloni 6394 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
| 12 | ordeq 6391 | . . . 4 ⊢ (𝐴 = On → (Ord 𝐴 ↔ Ord On)) | |
| 13 | 4, 12 | mpbiri 258 | . . 3 ⊢ (𝐴 = On → Ord 𝐴) |
| 14 | 11, 13 | jaoi 858 | . 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → Ord 𝐴) |
| 15 | 10, 14 | impbii 209 | 1 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∨ wo 848 ∨ w3o 1086 = wceq 1540 ∈ wcel 2108 Vcvv 3480 Ord word 6383 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: ordsson 7803 ssonprc 7807 ordunisuc 7852 orduninsuc 7864 limomss 7892 omon 7899 limom 7903 tfrlem14 8431 tfr2b 8436 unialeph 10141 ordtoplem 36436 ordcmp 36448 onsupnmax 43240 dflim5 43342 |
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