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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. (Contributed by NM, 1-Nov-2003.) |
Ref | Expression |
---|---|
epweon | ⊢ E We On |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | onfr 6198 | . 2 ⊢ E Fr On | |
2 | eloni 6169 | . . . 4 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
3 | eloni 6169 | . . . 4 ⊢ (𝑦 ∈ On → Ord 𝑦) | |
4 | ordtri3or 6191 | . . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | |
5 | epel 5433 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
6 | biid 264 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
7 | epel 5433 | . . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
8 | 5, 6, 7 | 3orbi123i 1153 | . . . . 5 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
9 | 4, 8 | sylibr 237 | . . . 4 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) |
10 | 2, 3, 9 | syl2an 598 | . . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) |
11 | 10 | rgen2 3168 | . 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) |
12 | dfwe2 7476 | . 2 ⊢ ( E We On ↔ ( E Fr On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))) | |
13 | 1, 11, 12 | mpbir2an 710 | 1 ⊢ E We On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 399 ∨ w3o 1083 ∈ wcel 2111 ∀wral 3106 class class class wbr 5030 E cep 5429 Fr wfr 5475 We wwe 5477 Ord word 6158 Oncon0 6159 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-ord 6162 df-on 6163 |
This theorem is referenced by: ordon 7478 omsinds 7580 onnseq 7964 dfrecs3 7992 tfr1ALT 8019 tfr2ALT 8020 tfr3ALT 8021 ordunifi 8752 ordtypelem8 8973 oismo 8988 cantnfcl 9114 leweon 9422 r0weon 9423 ac10ct 9445 dfac12lem2 9555 cflim2 9674 cofsmo 9680 hsmexlem1 9837 smobeth 9997 gruina 10229 ltsopi 10299 dford5 33070 finminlem 33779 dnwech 39992 aomclem4 40001 |
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