Theorem orduninsuc 7374

Description: An ordinal equal to its union is not a successor. (Contributed by NM, 18-Feb-2004.)

Assertion

Ref	Expression
orduninsuc	⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))

Distinct variable group:   𝑥,𝐴

Proof of Theorem orduninsuc

Step	Hyp	Ref	Expression
1		ordeleqon 7319	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		id 22	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → 𝐴 = if(𝐴 ∈ On, 𝐴, ∅))
3		unieq 4720	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ∪ 𝐴 = ∪ if(𝐴 ∈ On, 𝐴, ∅))
4	2, 3	eqeq12d 2793	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = ∪ 𝐴 ↔ if(𝐴 ∈ On, 𝐴, ∅) = ∪ if(𝐴 ∈ On, 𝐴, ∅)))
5		eqeq1 2782	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = suc 𝑥 ↔ if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
6	5	rexbidv 3242	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
7	6	notbid 310	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
8	4, 7	bibi12d 338	. . . 4 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ((𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (if(𝐴 ∈ On, 𝐴, ∅) = ∪ if(𝐴 ∈ On, 𝐴, ∅) ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥)))
9		0elon 6082	. . . . . 6 ⊢ ∅ ∈ On
10	9	elimel 4417	. . . . 5 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On
11	10	onuninsuci 7371	. . . 4 ⊢ (if(𝐴 ∈ On, 𝐴, ∅) = ∪ if(𝐴 ∈ On, 𝐴, ∅) ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥)
12	8, 11	dedth 4406	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
13		unon 7362	. . . . . 6 ⊢ ∪ On = On
14	13	eqcomi 2787	. . . . 5 ⊢ On = ∪ On
15		onprc 7315	. . . . . . . 8 ⊢ ¬ On ∈ V
16		vex 3418	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17	16	sucex 7342	. . . . . . . . 9 ⊢ suc 𝑥 ∈ V
18		eleq1 2853	. . . . . . . . 9 ⊢ (On = suc 𝑥 → (On ∈ V ↔ suc 𝑥 ∈ V))
19	17, 18	mpbiri 250	. . . . . . . 8 ⊢ (On = suc 𝑥 → On ∈ V)
20	15, 19	mto 189	. . . . . . 7 ⊢ ¬ On = suc 𝑥
21	20	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ On → ¬ On = suc 𝑥)
22	21	nrex 3214	. . . . 5 ⊢ ¬ ∃𝑥 ∈ On On = suc 𝑥
23	14, 22	2th 256	. . . 4 ⊢ (On = ∪ On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)
24		id 22	. . . . . 6 ⊢ (𝐴 = On → 𝐴 = On)
25		unieq 4720	. . . . . 6 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On)
26	24, 25	eqeq12d 2793	. . . . 5 ⊢ (𝐴 = On → (𝐴 = ∪ 𝐴 ↔ On = ∪ On))
27		eqeq1 2782	. . . . . . 7 ⊢ (𝐴 = On → (𝐴 = suc 𝑥 ↔ On = suc 𝑥))
28	27	rexbidv 3242	. . . . . 6 ⊢ (𝐴 = On → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On On = suc 𝑥))
29	28	notbid 310	. . . . 5 ⊢ (𝐴 = On → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥))
30	26, 29	bibi12d 338	. . . 4 ⊢ (𝐴 = On → ((𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (On = ∪ On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)))
31	23, 30	mpbiri 250	. . 3 ⊢ (𝐴 = On → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
32	12, 31	jaoi 843	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
33	1, 32	sylbi 209	1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∨ wo 833   = wceq 1507   ∈ wcel 2050  ∃wrex 3089  Vcvv 3415  ∅c0 4178  ifcif 4350  ∪ cuni 4712  Ord word 6028  Oncon0 6029  suc csuc 6031

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186  ax-un 7279

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-pss 3845  df-nul 4179  df-if 4351  df-pw 4424  df-sn 4442  df-pr 4444  df-tp 4446  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-tr 5031  df-eprel 5317  df-po 5326  df-so 5327  df-fr 5366  df-we 5368  df-ord 6032  df-on 6033  df-suc 6035

This theorem is referenced by:  ordunisuc2  7375  ordzsl  7376  dflim3  7378  nnsuc  7413