Theorem epweon 7775

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 7734, see epweonALT 7776. (Contributed by NM, 1-Nov-2003.) Avoid ax-un 7734. (Revised by BTernaryTau, 30-Nov-2024.)

Assertion

Ref	Expression
epweon	⊢ E We On

Proof of Theorem epweon

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

Step	Hyp	Ref	Expression
1		onfr 6402	. 2 ⊢ E Fr On
2		df-po 5571	. . . 4 ⊢ ( E Po On ↔ ∀𝑥 ∈ On ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
3		eloni 6372	. . . . . . . . 9 ⊢ (𝑥 ∈ On → Ord 𝑥)
4		ordirr 6380	. . . . . . . . 9 ⊢ (Ord 𝑥 → ¬ 𝑥 ∈ 𝑥)
5	3, 4	syl 18	. . . . . . . 8 ⊢ (𝑥 ∈ On → ¬ 𝑥 ∈ 𝑥)
6		epel 5566	. . . . . . . 8 ⊢ (𝑥 E 𝑥 ↔ 𝑥 ∈ 𝑥)
7	5, 6	sylnibr 332	. . . . . . 7 ⊢ (𝑥 ∈ On → ¬ 𝑥 E 𝑥)
8		ontr1 6410	. . . . . . . 8 ⊢ (𝑧 ∈ On → ((𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧) → 𝑥 ∈ 𝑧))
9		epel 5566	. . . . . . . . 9 ⊢ (𝑥 E 𝑦 ↔ 𝑥 ∈ 𝑦)
10		epel 5566	. . . . . . . . 9 ⊢ (𝑦 E 𝑧 ↔ 𝑦 ∈ 𝑧)
11	9, 10	anbi12i 639	. . . . . . . 8 ⊢ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) ↔ (𝑥 ∈ 𝑦 ∧ 𝑦 ∈ 𝑧))
12		epel 5566	. . . . . . . 8 ⊢ (𝑥 E 𝑧 ↔ 𝑥 ∈ 𝑧)
13	8, 11, 12	3imtr4g 299	. . . . . . 7 ⊢ (𝑧 ∈ On → ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧))
14	7, 13	anim12i 624	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑧 ∈ On) → (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
15	14	ralrimiva 3157	. . . . 5 ⊢ (𝑥 ∈ On → ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
16	15	ralrimivw 3161	. . . 4 ⊢ (𝑥 ∈ On → ∀𝑦 ∈ On ∀𝑧 ∈ On (¬ 𝑥 E 𝑥 ∧ ((𝑥 E 𝑦 ∧ 𝑦 E 𝑧) → 𝑥 E 𝑧)))
17	2, 16	mprgbir 3086	. . 3 ⊢ E Po On
18		eloni 6372	. . . . 5 ⊢ (𝑦 ∈ On → Ord 𝑦)
19		ordtri3or 6395	. . . . . 6 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
20		biid 264	. . . . . . 7 ⊢ (𝑥 = 𝑦 ↔ 𝑥 = 𝑦)
21		epel 5566	. . . . . . 7 ⊢ (𝑦 E 𝑥 ↔ 𝑦 ∈ 𝑥)
22	9, 20, 21	3orbi123i 1174	. . . . . 6 ⊢ ((𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥) ↔ (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
23	19, 22	sylibr 237	. . . . 5 ⊢ ((Ord 𝑥 ∧ Ord 𝑦) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
24	3, 18, 23	syl2an 607	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥))
25	24	rgen2 3205	. . 3 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)
26		df-so 5572	. . 3 ⊢ ( E Or On ↔ ( E Po On ∧ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 E 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 E 𝑥)))
27	17, 25, 26	mpbir2an 723	. 2 ⊢ E Or On
28		df-we 5618	. 2 ⊢ ( E We On ↔ ( E Fr On ∧ E Or On))
29	1, 27, 28	mpbir2an 723	1 ⊢ E We On

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ w3o 1102   ∈ wcel 2143  ∀wral 3079   class class class wbr 5110   E cep 5562   Po wpo 5569   Or wor 5570   Fr wfr 5613   We wwe 5615  Ord word 6361  Oncon0 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4489  df-pw 4565  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-tr 5220  df-eprel 5563  df-po 5571  df-so 5572  df-fr 5616  df-we 5618  df-ord 6365  df-on 6366

This theorem is referenced by:  ordon  7777  dford5  7784  omsinds  7884  onnseq  8332  dfrecs3  8360  tfr1ALT  8388  tfr2ALT  8389  tfr3ALT  8390  on2recsfn  8654  on2recsov  8655  on2ind  8656  on3ind  8657  ordunifi  9251  ordtypelem8  9488  oismo  9503  cantnfcl  9637  leweon  9996  r0weon  9997  ac10ct  10019  dfac12lem2  10129  cflim2  10248  cofsmo  10254  hsmexlem1  10411  smobeth  10572  gruina  10804  ltsopi  10874  onswe  28446  finminlem  36810  dnwech  43758  aomclem4  43767  onsupuni  43939  oninfint  43946  epsoon  43963  epirron  43964  oneptr  43965  oaun3lem1  44084