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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 6223 | . 2 ⊢ E Fr On | |
2 | eloni 6194 | . . . 4 ⊢ (𝑥 ∈ On → Ord 𝑥) | |
3 | eloni 6194 | . . . 4 ⊢ (𝑦 ∈ On → Ord 𝑦) | |
4 | ordtri3or 6216 | . . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) | |
5 | epel 5462 | . . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦) | |
6 | biid 262 | . . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦) | |
7 | epel 5462 | . . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥) | |
8 | 5, 6, 7 | 3orbi123i 1148 | . . . . 5 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) |
9 | 4, 8 | sylibr 235 | . . . 4 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) |
10 | 2, 3, 9 | syl2an 595 | . . 3 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)) |
11 | 10 | rgen2 3200 | . 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) |
12 | dfwe2 7485 | . 2 ⊢ ( E We On ↔ ( E Fr On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))) | |
13 | 1, 11, 12 | mpbir2an 707 | 1 ⊢ E We On |
Colors of variables: wff setvar class |
Syntax hints: ∧ wa 396 ∨ w3o 1078 ∈ wcel 2105 ∀wral 3135 class class class wbr 5057 E cep 5457 Fr wfr 5504 We wwe 5506 Ord word 6183 Oncon0 6184 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 |
This theorem is referenced by: ordon 7487 omsinds 7589 onnseq 7970 dfrecs3 7998 tfr1ALT 8025 tfr2ALT 8026 tfr3ALT 8027 ordunifi 8756 ordtypelem8 8977 oismo 8992 cantnfcl 9118 leweon 9425 r0weon 9426 ac10ct 9448 dfac12lem2 9558 cflim2 9673 cofsmo 9679 hsmexlem1 9836 smobeth 9996 gruina 10228 ltsopi 10298 dford5 32854 finminlem 33563 dnwech 39526 aomclem4 39535 |
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