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Mirrors > Home > MPE Home > Th. List > dford2 | Structured version Visualization version GIF version |
Description: Assuming ax-reg 9586, an ordinal is a transitive class on which inclusion satisfies trichotomy. (Contributed by Scott Fenton, 27-Oct-2010.) |
Ref | Expression |
---|---|
dford2 | ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ord 6367 | . 2 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
2 | zfregfr 9599 | . . . . 5 ⊢ E Fr 𝐴 | |
3 | dfwe2 7760 | . . . . 5 ⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))) | |
4 | 2, 3 | mpbiran 707 | . . . 4 ⊢ ( E We 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) |
5 | epel 5583 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
6 | biid 260 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
7 | epel 5583 | . . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
8 | 5, 6, 7 | 3orbi123i 1156 | . . . . 5 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
9 | 8 | 2ralbii 3128 | . . . 4 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
10 | 4, 9 | bitri 274 | . . 3 ⊢ ( E We 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
11 | 10 | anbi2i 623 | . 2 ⊢ ((Tr 𝐴 ∧ E We 𝐴) ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))) |
12 | 1, 11 | bitri 274 | 1 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 396 ∨ w3o 1086 ∀wral 3061 class class class wbr 5148 Tr wtr 5265 E cep 5579 Fr wfr 5628 We wwe 5630 Ord word 6363 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 ax-un 7724 ax-reg 9586 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-pw 4604 df-sn 4629 df-pr 4631 df-tp 4633 df-op 4635 df-uni 4909 df-br 5149 df-opab 5211 df-eprel 5580 df-po 5588 df-so 5589 df-fr 5631 df-we 5633 df-ord 6367 |
This theorem is referenced by: ordelordALT 43288 ordelordALTVD 43618 |
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