Theorem orduninsuc 7560

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 7505	. 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2		id 22	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → 𝐴 = if(𝐴 ∈ On, 𝐴, ∅))
3		unieq 4851	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ∪ 𝐴 = ∪ if(𝐴 ∈ On, 𝐴, ∅))
4	2, 3	eqeq12d 2839	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = ∪ 𝐴 ↔ if(𝐴 ∈ On, 𝐴, ∅) = ∪ if(𝐴 ∈ On, 𝐴, ∅)))
5		eqeq1 2827	. . . . . . 7 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 = suc 𝑥 ↔ if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
6	5	rexbidv 3299	. . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
7	6	notbid 320	. . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥))
8	4, 7	bibi12d 348	. . . 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 6246	. . . . . 6 ⊢ ∅ ∈ On
10	9	elimel 4536	. . . . 5 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On
11	10	onuninsuci 7557	. . . 4 ⊢ (if(𝐴 ∈ On, 𝐴, ∅) = ∪ if(𝐴 ∈ On, 𝐴, ∅) ↔ ¬ ∃𝑥 ∈ On if(𝐴 ∈ On, 𝐴, ∅) = suc 𝑥)
12	8, 11	dedth 4525	. . 3 ⊢ (𝐴 ∈ On → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
13		unon 7548	. . . . . 6 ⊢ ∪ On = On
14	13	eqcomi 2832	. . . . 5 ⊢ On = ∪ On
15		onprc 7501	. . . . . . . 8 ⊢ ¬ On ∈ V
16		vex 3499	. . . . . . . . . 10 ⊢ 𝑥 ∈ V
17	16	sucex 7528	. . . . . . . . 9 ⊢ suc 𝑥 ∈ V
18		eleq1 2902	. . . . . . . . 9 ⊢ (On = suc 𝑥 → (On ∈ V ↔ suc 𝑥 ∈ V))
19	17, 18	mpbiri 260	. . . . . . . 8 ⊢ (On = suc 𝑥 → On ∈ V)
20	15, 19	mto 199	. . . . . . 7 ⊢ ¬ On = suc 𝑥
21	20	a1i 11	. . . . . 6 ⊢ (𝑥 ∈ On → ¬ On = suc 𝑥)
22	21	nrex 3271	. . . . 5 ⊢ ¬ ∃𝑥 ∈ On On = suc 𝑥
23	14, 22	2th 266	. . . 4 ⊢ (On = ∪ On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)
24		id 22	. . . . . 6 ⊢ (𝐴 = On → 𝐴 = On)
25		unieq 4851	. . . . . 6 ⊢ (𝐴 = On → ∪ 𝐴 = ∪ On)
26	24, 25	eqeq12d 2839	. . . . 5 ⊢ (𝐴 = On → (𝐴 = ∪ 𝐴 ↔ On = ∪ On))
27		eqeq1 2827	. . . . . . 7 ⊢ (𝐴 = On → (𝐴 = suc 𝑥 ↔ On = suc 𝑥))
28	27	rexbidv 3299	. . . . . 6 ⊢ (𝐴 = On → (∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ∃𝑥 ∈ On On = suc 𝑥))
29	28	notbid 320	. . . . 5 ⊢ (𝐴 = On → (¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥 ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥))
30	26, 29	bibi12d 348	. . . 4 ⊢ (𝐴 = On → ((𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥) ↔ (On = ∪ On ↔ ¬ ∃𝑥 ∈ On On = suc 𝑥)))
31	23, 30	mpbiri 260	. . 3 ⊢ (𝐴 = On → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
32	12, 31	jaoi 853	. 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))
33	1, 32	sylbi 219	1 ⊢ (Ord 𝐴 → (𝐴 = ∪ 𝐴 ↔ ¬ ∃𝑥 ∈ On 𝐴 = suc 𝑥))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∨ wo 843   = wceq 1537   ∈ wcel 2114  ∃wrex 3141  Vcvv 3496  ∅c0 4293  ifcif 4469  ∪ cuni 4840  Ord word 6192  Oncon0 6193  suc csuc 6195

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ne 3019  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-sbc 3775  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-pss 3956  df-nul 4294  df-if 4470  df-pw 4543  df-sn 4570  df-pr 4572  df-tp 4574  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-tr 5175  df-eprel 5467  df-po 5476  df-so 5477  df-fr 5516  df-we 5518  df-ord 6196  df-on 6197  df-suc 6199

This theorem is referenced by:  ordunisuc2  7561  ordzsl  7562  dflim3  7564  nnsuc  7599