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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 7725, see epweonALT 7763. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7725. (Revised by BTernaryTau, 30-Nov-2024.) |
Ref | Expression |
---|---|
epweon | ⊢ E We On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onfr 6404 | . 2 ⊢ E Fr On | |
2 | df-po 5589 | . . . 4 ⊢ ( E Po On ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))) | |
3 | eloni 6375 | . . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
4 | ordirr 6383 | . . . . . . . . 9 ⊢ (Ord 𝑥 → ¬ 𝑥 ∈ 𝑥) | |
5 | 3, 4 | syl 17 | . . . . . . . 8 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ 𝑥) |
6 | epel 5584 | . . . . . . . 8 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥) | |
7 | 5, 6 | sylnibr 329 | . . . . . . 7 ⊢ (𝑥 ∈ On → ¬ 𝑥 E 𝑥) |
8 | ontr1 6411 | . . . . . . . 8 ⊢ (𝑧 ∈ On → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧)) | |
9 | epel 5584 | . . . . . . . . 9 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
10 | epel 5584 | . . . . . . . . 9 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧) | |
11 | 9, 10 | anbi12i 628 | . . . . . . . 8 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧)) |
12 | epel 5584 | . . . . . . . 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 614 | . . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))) |
15 | 14 | ralrimiva 3147 | . . . . 5 ⊢ (𝑥 ∈ On → ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))) |
16 | 15 | ralrimivw 3151 | . . . 4 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))) |
17 | 2, 16 | mprgbir 3069 | . . 3 ⊢ E Po On |
18 | eloni 6375 | . . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦) | |
19 | ordtri3or 6397 | . . . . . 6 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | |
20 | biid 261 | . . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
21 | epel 5584 | . . . . . . 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 233 | . . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) |
24 | 3, 18, 23 | syl2an 597 | . . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) |
25 | 24 | rgen2 3198 | . . 3 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) |
26 | df-so 5590 | . . 3 ⊢ ( E Or On ↔ ( E Po On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))) | |
27 | 17, 25, 26 | mpbir2an 710 | . 2 ⊢ E Or On |
28 | df-we 5634 | . 2 ⊢ ( E We On ↔ ( E Fr On ∧ E Or On)) | |
29 | 1, 27, 28 | mpbir2an 710 | 1 ⊢ E We On |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 ∨ w3o 1087 ∈ wcel 2107 ∀wral 3062 class class class wbr 5149 E cep 5580 Po wpo 5587 Or wor 5588 Fr wfr 5629 We wwe 5631 Ord word 6364 Oncon0 6365 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-br 5150 df-opab 5212 df-tr 5267 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-ord 6368 df-on 6369 |
This theorem is referenced by: ordon 7764 dford5 7771 omsinds 7876 omsindsOLD 7877 onnseq 8344 dfrecs3 8372 dfrecs3OLD 8373 tfr1ALT 8400 tfr2ALT 8401 tfr3ALT 8402 on2recsfn 8666 on2recsov 8667 on2ind 8668 on3ind 8669 ordunifi 9293 ordtypelem8 9520 oismo 9535 cantnfcl 9662 leweon 10006 r0weon 10007 ac10ct 10029 dfac12lem2 10139 cflim2 10258 cofsmo 10264 hsmexlem1 10421 smobeth 10581 gruina 10813 ltsopi 10883 finminlem 35251 dnwech 41838 aomclem4 41847 onsupuni 42026 oninfint 42033 epsoon 42050 epirron 42051 oneptr 42052 oaun3lem1 42172 |
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