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| Mirrors > Home > MPE Home > Th. List > epweon | Structured version Visualization version GIF version | ||
| Description: The membership relation well-orders the class of ordinal numbers. This proof does not require the axiom of regularity. Proposition 4.8(g) of [Mendelson] p. 244. For a shorter proof requiring ax-un 7755, see epweonALT 7796. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7755. (Revised by BTernaryTau, 30-Nov-2024.) |
| Ref | Expression |
|---|---|
| epweon | ⊢ E We On |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | onfr 6423 | . 2 ⊢ E Fr On | |
| 2 | df-po 5592 | . . . 4 ⊢ ( E Po On ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))) | |
| 3 | eloni 6394 | . . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
| 4 | ordirr 6402 | . . . . . . . . 9 ⊢ (Ord 𝑥 → ¬ 𝑥 ∈ 𝑥) | |
| 5 | 3, 4 | syl 17 | . . . . . . . 8 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ 𝑥) |
| 6 | epel 5587 | . . . . . . . 8 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥) | |
| 7 | 5, 6 | sylnibr 329 | . . . . . . 7 ⊢ (𝑥 ∈ On → ¬ 𝑥 E 𝑥) |
| 8 | ontr1 6430 | . . . . . . . 8 ⊢ (𝑧 ∈ On → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) | |
| 9 | epel 5587 | . . . . . . . . 9 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
| 10 | epel 5587 | . . . . . . . . 9 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
| 11 | 9, 10 | anbi12i 628 | . . . . . . . 8 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧)) |
| 12 | epel 5587 | . . . . . . . 8 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧) | |
| 13 | 8, 11, 12 | 3imtr4g 296 | . . . . . . 7 ⊢ (𝑧 ∈ On → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)) |
| 14 | 7, 13 | anim12i 613 | . . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))) |
| 15 | 14 | ralrimiva 3146 | . . . . 5 ⊢ (𝑥 ∈ On → ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))) |
| 16 | 15 | ralrimivw 3150 | . . . 4 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))) |
| 17 | 2, 16 | mprgbir 3068 | . . 3 ⊢ E Po On |
| 18 | eloni 6394 | . . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦) | |
| 19 | ordtri3or 6416 | . . . . . 6 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | |
| 20 | biid 261 | . . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
| 21 | epel 5587 | . . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
| 22 | 9, 20, 21 | 3orbi123i 1157 | . . . . . 6 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
| 23 | 19, 22 | sylibr 234 | . . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) |
| 24 | 3, 18, 23 | syl2an 596 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) |
| 25 | 24 | rgen2 3199 | . . 3 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) |
| 26 | df-so 5593 | . . 3 ⊢ ( E Or On ↔ ( E Po On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))) | |
| 27 | 17, 25, 26 | mpbir2an 711 | . 2 ⊢ E Or On |
| 28 | df-we 5639 | . 2 ⊢ ( E We On ↔ ( E Fr On ∧ E Or On)) | |
| 29 | 1, 27, 28 | mpbir2an 711 | 1 ⊢ E We On |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∨ w3o 1086 ∈ wcel 2108 ∀wral 3061 class class class wbr 5143 E cep 5583 Po wpo 5590 Or wor 5591 Fr wfr 5634 We wwe 5636 Ord word 6383 Oncon0 6384 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-ord 6387 df-on 6388 |
| This theorem is referenced by: ordon 7797 dford5 7804 omsinds 7908 onnseq 8384 dfrecs3 8412 dfrecs3OLD 8413 tfr1ALT 8440 tfr2ALT 8441 tfr3ALT 8442 on2recsfn 8705 on2recsov 8706 on2ind 8707 on3ind 8708 ordunifi 9326 ordtypelem8 9565 oismo 9580 cantnfcl 9707 leweon 10051 r0weon 10052 ac10ct 10074 dfac12lem2 10185 cflim2 10303 cofsmo 10309 hsmexlem1 10466 smobeth 10626 gruina 10858 ltsopi 10928 finminlem 36319 dnwech 43060 aomclem4 43069 onsupuni 43241 oninfint 43248 epsoon 43265 epirron 43266 oneptr 43267 oaun3lem1 43387 |
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