![]() |
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 4346 | . 2 ⊢ (Ord 𝐴 → E Fr 𝐴) | |
2 | ordelord 4165 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝑧 ∈ 𝐴) → Ord 𝑧) | |
3 | 2 | 3ad2antr3 1106 | . . . 4 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → Ord 𝑧) |
4 | ordtr1 4172 | . . . . 5 ⊢ (Ord 𝑧 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) | |
5 | epel 4076 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
6 | epel 4076 | . . . . . 6 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
7 | 5, 6 | anbi12i 448 | . . . . 5 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧)) |
8 | epel 4076 | . . . . 5 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧) | |
9 | 4, 7, 8 | 3imtr4g 203 | . . . 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 2450 | . 2 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)) |
12 | df-wetr 4118 | . 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 408 | 1 ⊢ (Ord 𝐴 → E We 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 ∧ w3a 920 ∈ wcel 1434 ∀wral 2353 class class class wbr 3806 E cep 4071 Fr wfr 4112 We wwe 4114 Ord word 4146 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3917 ax-pow 3969 ax-pr 3993 ax-setind 4309 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-v 2613 df-un 2987 df-in 2989 df-ss 2996 df-pw 3403 df-sn 3423 df-pr 3424 df-op 3426 df-uni 3623 df-br 3807 df-opab 3861 df-tr 3897 df-eprel 4073 df-frfor 4115 df-frind 4116 df-wetr 4118 df-iord 4150 |
This theorem is referenced by: nnwetri 6462 |
Copyright terms: Public domain | W3C validator |