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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 4299 | . 2 ⊢ (Ord 𝐴 → E Fr 𝐴) | |
2 | ordelord 4118 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝑧 ∈ 𝐴) → Ord 𝑧) | |
3 | 2 | 3ad2antr3 1071 | . . . 4 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → Ord 𝑧) |
4 | ordtr1 4125 | . . . . 5 ⊢ (Ord 𝑧 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) | |
5 | epel 4029 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
6 | epel 4029 | . . . . . 6 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
7 | 5, 6 | anbi12i 433 | . . . . 5 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧)) |
8 | epel 4029 | . . . . 5 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧) | |
9 | 4, 7, 8 | 3imtr4g 194 | . . . 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 2402 | . 2 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)) |
12 | df-wetr 4071 | . 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 394 | 1 ⊢ (Ord 𝐴 → E We 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∧ w3a 885 ∈ wcel 1393 ∀wral 2306 class class class wbr 3764 E cep 4024 Fr wfr 4065 We wwe 4067 Ord word 4099 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-sep 3875 ax-pow 3927 ax-pr 3944 ax-setind 4262 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ral 2311 df-rex 2312 df-v 2559 df-un 2922 df-in 2924 df-ss 2931 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-br 3765 df-opab 3819 df-tr 3855 df-eprel 4026 df-frfor 4068 df-frind 4069 df-wetr 4071 df-iord 4103 |
This theorem is referenced by: nnwetri 6354 |
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