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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 4497 | . 2 ⊢ (Ord 𝐴 → E Fr 𝐴) | |
2 | ordelord 4311 | . . . . 5 ⊢ ((Ord 𝐴 ∧ 𝑧 ∈ 𝐴) → Ord 𝑧) | |
3 | 2 | 3ad2antr3 1149 | . . . 4 ⊢ ((Ord 𝐴 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → Ord 𝑧) |
4 | ordtr1 4318 | . . . . 5 ⊢ (Ord 𝑧 → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) | |
5 | epel 4222 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
6 | epel 4222 | . . . . . 6 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
7 | 5, 6 | anbi12i 456 | . . . . 5 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧)) |
8 | epel 4222 | . . . . 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 2518 | . 2 ⊢ (Ord 𝐴 → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)) |
12 | df-wetr 4264 | . 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 414 | 1 ⊢ (Ord 𝐴 → E We 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∧ w3a 963 ∈ wcel 1481 ∀wral 2417 class class class wbr 3937 E cep 4217 Fr wfr 4258 We wwe 4260 Ord word 4292 |
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 699 ax-5 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-pow 4106 ax-pr 4139 ax-setind 4460 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1335 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ral 2422 df-rex 2423 df-v 2691 df-un 3080 df-in 3082 df-ss 3089 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-br 3938 df-opab 3998 df-tr 4035 df-eprel 4219 df-frfor 4261 df-frind 4262 df-wetr 4264 df-iord 4296 |
This theorem is referenced by: nnwetri 6812 |
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