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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 6876 | . . . 4 ⊢ ¬ On ∈ V | |
2 | elex 3185 | . . . 4 ⊢ (On ∈ 𝐴 → On ∈ V) | |
3 | 1, 2 | mto 187 | . . 3 ⊢ ¬ On ∈ 𝐴 |
4 | ordon 6874 | . . . . . 6 ⊢ Ord On | |
5 | ordtri3or 5672 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴)) | |
6 | 4, 5 | mpan2 703 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴)) |
7 | df-3or 1032 | . . . . 5 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴) ↔ ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴)) | |
8 | 6, 7 | sylib 207 | . . . 4 ⊢ (Ord 𝐴 → ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴)) |
9 | 8 | ord 391 | . . 3 ⊢ (Ord 𝐴 → (¬ (𝐴 ∈ On ∨ 𝐴 = On) → On ∈ 𝐴)) |
10 | 3, 9 | mt3i 140 | . 2 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On)) |
11 | eloni 5650 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
12 | ordeq 5647 | . . . 4 ⊢ (𝐴 = On → (Ord 𝐴 ↔ Ord On)) | |
13 | 4, 12 | mpbiri 247 | . . 3 ⊢ (𝐴 = On → Ord 𝐴) |
14 | 11, 13 | jaoi 393 | . 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → Ord 𝐴) |
15 | 10, 14 | impbii 198 | 1 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 195 ∨ wo 382 ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 Vcvv 3173 Ord word 5639 Oncon0 5640 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-tr 4681 df-eprel 4949 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-ord 5643 df-on 5644 |
This theorem is referenced by: ordsson 6881 ssonprc 6884 ordunisuc 6924 orduninsuc 6935 limomss 6962 omon 6968 limom 6972 tfrlem14 7374 tfr2b 7379 unialeph 8807 ordtoplem 31604 ordcmp 31616 |
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