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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 4459 | . 2 ⊢ (Ord 𝐴 → E Fr 𝐴) | |
2 | ordelord 4273 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝑧 ∈ 𝐴) → Ord 𝑧) | |
3 | 2 | 3ad2antr3 1133 | . . . 4 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → Ord 𝑧) |
4 | ordtr1 4280 | . . . . 5 ⊢ (Ord 𝑧 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) | |
5 | epel 4184 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
6 | epel 4184 | . . . . . 6 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
7 | 5, 6 | anbi12i 455 | . . . . 5 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧)) |
8 | epel 4184 | . . . . 5 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧) | |
9 | 4, 7, 8 | 3imtr4g 204 | . . . 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 2492 | . 2 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)) |
12 | df-wetr 4226 | . 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 413 | 1 ⊢ (Ord 𝐴 → E We 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 947 ∈ wcel 1465 ∀wral 2393 class class class wbr 3899 E cep 4179 Fr wfr 4220 We wwe 4222 Ord word 4254 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 ax-setind 4422 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-tr 3997 df-eprel 4181 df-frfor 4223 df-frind 4224 df-wetr 4226 df-iord 4258 |
This theorem is referenced by: nnwetri 6772 |
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