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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. Proposition 4.8(g) of [Mendelson] p. 244. (Contributed by NM, 1-Nov-2003.) |
Ref | Expression |
---|---|
epweon | ⊢ E We On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onfr 6006 | . 2 ⊢ E Fr On | |
2 | eloni 5977 | . . . 4 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
3 | eloni 5977 | . . . 4 ⊢ (𝑦 ∈ On → Ord 𝑦) | |
4 | ordtri3or 5999 | . . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | |
5 | epel 5260 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
6 | biid 253 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
7 | epel 5260 | . . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
8 | 5, 6, 7 | 3orbi123i 1199 | . . . . 5 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
9 | 4, 8 | sylibr 226 | . . . 4 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) |
10 | 2, 3, 9 | syl2an 589 | . . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) |
11 | 10 | rgen2a 3186 | . 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) |
12 | dfwe2 7247 | . 2 ⊢ ( E We On ↔ ( E Fr On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))) | |
13 | 1, 11, 12 | mpbir2an 702 | 1 ⊢ E We On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 386 ∨ w3o 1110 ∈ wcel 2164 ∀wral 3117 class class class wbr 4875 E cep 5256 Fr wfr 5302 We wwe 5304 Ord word 5966 Oncon0 5967 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pr 5129 ax-un 7214 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3or 1112 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4147 df-if 4309 df-sn 4400 df-pr 4402 df-tp 4404 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-tr 4978 df-eprel 5257 df-po 5265 df-so 5266 df-fr 5305 df-we 5307 df-ord 5970 df-on 5971 |
This theorem is referenced by: ordon 7249 omsinds 7350 onnseq 7712 dfrecs3 7740 tfr1ALT 7767 tfr2ALT 7768 tfr3ALT 7769 ordunifi 8485 ordtypelem8 8706 oismo 8721 cantnfcl 8848 leweon 9154 r0weon 9155 ac10ct 9177 dfac12lem2 9288 cflim2 9407 cofsmo 9413 hsmexlem1 9570 smobeth 9730 gruina 9962 ltsopi 10032 dford5 32148 finminlem 32846 dnwech 38460 aomclem4 38469 |
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