Theorem sucprcreg 4520

Description: A class is equal to its successor iff it is a proper class (assuming the Axiom of Set Induction). (Contributed by NM, 9-Jul-2004.)

Assertion

Ref	Expression
sucprcreg	⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Proof of Theorem sucprcreg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sucprc 4384	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 4512	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		nfv 1515	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐴
4		eleq1 2227	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
5	3, 4	ceqsalg 2749	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴) ↔ 𝐴 ∈ 𝐴))
6	2, 5	mtbiri 665	. . 3 ⊢ (𝐴 ∈ V → ¬ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴))
7		velsn 3587	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8		olc 701	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
9		elun 3258	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
10		ssid 3157	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
11		df-suc 4343	. . . . . . . . . . 11 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
12	11	eqeq1i 2172	. . . . . . . . . 10 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
13		sseq1 3160	. . . . . . . . . 10 ⊢ ((𝐴 ∪ {𝐴}) = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
14	12, 13	sylbi 120	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
15	10, 14	mpbiri 167	. . . . . . . 8 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∪ {𝐴}) ⊆ 𝐴)
16	15	sseld 3136	. . . . . . 7 ⊢ (suc 𝐴 = 𝐴 → (𝑥 ∈ (𝐴 ∪ {𝐴}) → 𝑥 ∈ 𝐴))
17	9, 16	syl5bir 152	. . . . . 6 ⊢ (suc 𝐴 = 𝐴 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) → 𝑥 ∈ 𝐴))
18	8, 17	syl5 32	. . . . 5 ⊢ (suc 𝐴 = 𝐴 → (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐴))
19	7, 18	syl5bir 152	. . . 4 ⊢ (suc 𝐴 = 𝐴 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐴))
20	19	alrimiv 1861	. . 3 ⊢ (suc 𝐴 = 𝐴 → ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴))
21	6, 20	nsyl3 616	. 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
22	1, 21	impbii 125	1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 104   ∨ wo 698  ∀wal 1340   = wceq 1342   ∈ wcel 2135  Vcvv 2721   ∪ cun 3109   ⊆ wss 3111  {csn 3570  suc csuc 4337

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-ext 2146  ax-setind 4508

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-ral 2447  df-v 2723  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-nul 3405  df-sn 3576  df-suc 4343

This theorem is referenced by: (None)