![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > ordwe | GIF version |
Description: Epsilon well-orders every ordinal. Proposition 7.4 of [TakeutiZaring] p. 36. (Contributed by NM, 3-Apr-1994.) |
Ref | Expression |
---|---|
ordwe | ⊢ (Ord 𝐴 → E We 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordfr 4574 | . 2 ⊢ (Ord 𝐴 → E Fr 𝐴) | |
2 | ordelord 4381 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝑧 ∈ 𝐴) → Ord 𝑧) | |
3 | 2 | 3ad2antr3 1164 | . . . 4 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → Ord 𝑧) |
4 | ordtr1 4388 | . . . . 5 ⊢ (Ord 𝑧 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) | |
5 | epel 4292 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
6 | epel 4292 | . . . . . 6 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
7 | 5, 6 | anbi12i 460 | . . . . 5 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧)) |
8 | epel 4292 | . . . . 5 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧) | |
9 | 4, 7, 8 | 3imtr4g 205 | . . . 4 ⊢ (Ord 𝑧 → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)) |
10 | 3, 9 | syl 14 | . . 3 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)) |
11 | 10 | ralrimivvva 2560 | . 2 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)) |
12 | df-wetr 4334 | . 2 ⊢ ( E We 𝐴 ↔ ( E Fr 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))) | |
13 | 1, 11, 12 | sylanbrc 417 | 1 ⊢ (Ord 𝐴 → E We 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∧ w3a 978 ∈ wcel 2148 ∀wral 2455 class class class wbr 4003 E cep 4287 Fr wfr 4328 We wwe 4330 Ord word 4362 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4121 ax-pow 4174 ax-pr 4209 ax-setind 4536 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2739 df-un 3133 df-in 3135 df-ss 3142 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-tr 4102 df-eprel 4289 df-frfor 4331 df-frind 4332 df-wetr 4334 df-iord 4366 |
This theorem is referenced by: nnwetri 6914 |
Copyright terms: Public domain | W3C validator |