Theorem epweon 7603

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.)

Assertion

Ref	Expression
epweon	⊢ E We On

Proof of Theorem epweon

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		onfr 6290	. 2 ⊢ E Fr On
2		eloni 6261	. . . 4 ⊢ (𝑥 ∈ On → Ord 𝑥)
3		eloni 6261	. . . 4 ⊢ (𝑦 ∈ On → Ord 𝑦)
4		ordtri3or 6283	. . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
5		epel 5489	. . . . . 6 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
6		biid 260	. . . . . 6 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
7		epel 5489	. . . . . 6 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
8	5, 6, 7	3orbi123i 1154	. . . . 5 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
9	4, 8	sylibr 233	. . . 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 3126	. 2 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)
12		dfwe2 7602	. 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

Syntax hints:   ∧ wa 395   ∨ w3o 1084   ∈ wcel 2108  ∀wral 3063   class class class wbr 5070   E cep 5485   Fr wfr 5532   We wwe 5534  Ord word 6250  Oncon0 6251

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-11 2156  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255

This theorem is referenced by:  ordon  7604  omsinds  7708  omsindsOLD  7709  onnseq  8146  dfrecs3  8174  dfrecs3OLD  8175  tfr1ALT  8202  tfr2ALT  8203  tfr3ALT  8204  ordunifi  8994  ordtypelem8  9214  oismo  9229  cantnfcl  9355  leweon  9698  r0weon  9699  ac10ct  9721  dfac12lem2  9831  cflim2  9950  cofsmo  9956  hsmexlem1  10113  smobeth  10273  gruina  10505  ltsopi  10575  dford5  33573  on2recsfn  33753  on2recsov  33754  on2ind  33755  on3ind  33756  finminlem  34434  dnwech  40789  aomclem4  40798