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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 7488 | . . . 4 ⊢ ¬ On ∈ V | |
2 | elex 3510 | . . . 4 ⊢ (On ∈ 𝐴 → On ∈ V) | |
3 | 1, 2 | mto 198 | . . 3 ⊢ ¬ On ∈ 𝐴 |
4 | ordon 7487 | . . . . . 6 ⊢ Ord On | |
5 | ordtri3or 6216 | . . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord On) → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴)) | |
6 | 4, 5 | mpan2 687 | . . . . 5 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴)) |
7 | df-3or 1080 | . . . . 5 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On ∨ On ∈ 𝐴) ↔ ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴)) | |
8 | 6, 7 | sylib 219 | . . . 4 ⊢ (Ord 𝐴 → ((𝐴 ∈ On ∨ 𝐴 = On) ∨ On ∈ 𝐴)) |
9 | 8 | ord 858 | . . 3 ⊢ (Ord 𝐴 → (¬ (𝐴 ∈ On ∨ 𝐴 = On) → On ∈ 𝐴)) |
10 | 3, 9 | mt3i 151 | . 2 ⊢ (Ord 𝐴 → (𝐴 ∈ On ∨ 𝐴 = On)) |
11 | eloni 6194 | . . 3 ⊢ (𝐴 ∈ On → Ord 𝐴) | |
12 | ordeq 6191 | . . . 4 ⊢ (𝐴 = On → (Ord 𝐴 ↔ Ord On)) | |
13 | 4, 12 | mpbiri 259 | . . 3 ⊢ (𝐴 = On → Ord 𝐴) |
14 | 11, 13 | jaoi 851 | . 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → Ord 𝐴) |
15 | 10, 14 | impbii 210 | 1 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∨ wo 841 ∨ w3o 1078 = wceq 1528 ∈ wcel 2105 Vcvv 3492 Ord word 6183 Oncon0 6184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 |
This theorem is referenced by: ordsson 7493 ssonprc 7496 ordunisuc 7536 orduninsuc 7547 limomss 7574 omon 7580 limom 7584 tfrlem14 8016 tfr2b 8021 unialeph 9515 ordtoplem 33680 ordcmp 33692 |
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