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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 7753, see epweonALT 7794. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7753. (Revised by BTernaryTau, 30-Nov-2024.) |
Ref | Expression |
---|---|
epweon | ⊢ E We On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onfr 6424 | . 2 ⊢ E Fr On | |
2 | df-po 5596 | . . . 4 ⊢ ( E Po On ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))) | |
3 | eloni 6395 | . . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
4 | ordirr 6403 | . . . . . . . . 9 ⊢ (Ord 𝑥 → ¬ 𝑥 ∈ 𝑥) | |
5 | 3, 4 | syl 17 | . . . . . . . 8 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ 𝑥) |
6 | epel 5591 | . . . . . . . 8 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥) | |
7 | 5, 6 | sylnibr 329 | . . . . . . 7 ⊢ (𝑥 ∈ On → ¬ 𝑥 E 𝑥) |
8 | ontr1 6431 | . . . . . . . 8 ⊢ (𝑧 ∈ On → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) | |
9 | epel 5591 | . . . . . . . . 9 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
10 | epel 5591 | . . . . . . . . 9 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
11 | 9, 10 | anbi12i 628 | . . . . . . . 8 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧)) |
12 | epel 5591 | . . . . . . . 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 3143 | . . . . 5 ⊢ (𝑥 ∈ On → ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))) |
16 | 15 | ralrimivw 3147 | . . . 4 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))) |
17 | 2, 16 | mprgbir 3065 | . . 3 ⊢ E Po On |
18 | eloni 6395 | . . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦) | |
19 | ordtri3or 6417 | . . . . . 6 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | |
20 | biid 261 | . . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
21 | epel 5591 | . . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
22 | 9, 20, 21 | 3orbi123i 1155 | . . . . . 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 3196 | . . 3 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) |
26 | df-so 5597 | . . 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 5642 | . 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 1085 ∈ wcel 2105 ∀wral 3058 class class class wbr 5147 E cep 5587 Po wpo 5594 Or wor 5595 Fr wfr 5637 We wwe 5639 Ord word 6384 Oncon0 6385 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-sb 2062 df-clab 2712 df-cleq 2726 df-clel 2813 df-ne 2938 df-ral 3059 df-rex 3068 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-pss 3982 df-nul 4339 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-tr 5265 df-eprel 5588 df-po 5596 df-so 5597 df-fr 5640 df-we 5642 df-ord 6388 df-on 6389 |
This theorem is referenced by: ordon 7795 dford5 7802 omsinds 7907 omsindsOLD 7908 onnseq 8382 dfrecs3 8410 dfrecs3OLD 8411 tfr1ALT 8438 tfr2ALT 8439 tfr3ALT 8440 on2recsfn 8703 on2recsov 8704 on2ind 8705 on3ind 8706 ordunifi 9323 ordtypelem8 9562 oismo 9577 cantnfcl 9704 leweon 10048 r0weon 10049 ac10ct 10071 dfac12lem2 10182 cflim2 10300 cofsmo 10306 hsmexlem1 10463 smobeth 10623 gruina 10855 ltsopi 10925 finminlem 36300 dnwech 43036 aomclem4 43045 onsupuni 43217 oninfint 43224 epsoon 43241 epirron 43242 oneptr 43243 oaun3lem1 43363 |
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