![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > dford5 | Structured version Visualization version GIF version |
Description: A class is ordinal iff it is a subclass of On and transitive. (Contributed by Scott Fenton, 21-Nov-2021.) |
Ref | Expression |
---|---|
dford5 | ⊢ (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordsson 7770 | . . 3 ⊢ (Ord 𝐴 → 𝐴 ⊆ On) | |
2 | ordtr 6379 | . . 3 ⊢ (Ord 𝐴 → Tr 𝐴) | |
3 | 1, 2 | jca 513 | . 2 ⊢ (Ord 𝐴 → (𝐴 ⊆ On ∧ Tr 𝐴)) |
4 | epweon 7762 | . . . 4 ⊢ E We On | |
5 | wess 5664 | . . . 4 ⊢ (𝐴 ⊆ On → ( E We On → E We 𝐴)) | |
6 | 4, 5 | mpi 20 | . . 3 ⊢ (𝐴 ⊆ On → E We 𝐴) |
7 | df-ord 6368 | . . . . 5 ⊢ (Ord 𝐴 ↔ (Tr 𝐴 ∧ E We 𝐴)) | |
8 | 7 | biimpri 227 | . . . 4 ⊢ ((Tr 𝐴 ∧ E We 𝐴) → Ord 𝐴) |
9 | 8 | ancoms 460 | . . 3 ⊢ (( E We 𝐴 ∧ Tr 𝐴) → Ord 𝐴) |
10 | 6, 9 | sylan 581 | . 2 ⊢ ((𝐴 ⊆ On ∧ Tr 𝐴) → Ord 𝐴) |
11 | 3, 10 | impbii 208 | 1 ⊢ (Ord 𝐴 ↔ (𝐴 ⊆ On ∧ Tr 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ⊆ wss 3949 Tr wtr 5266 E cep 5580 We wwe 5631 Ord word 6364 Oncon0 6365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-ord 6368 df-on 6369 |
This theorem is referenced by: nosupno 27206 noinfno 27221 nadd2rabord 42135 nadd1rabord 42139 |
Copyright terms: Public domain | W3C validator |