Theorem sucprcreg 4641

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 4503	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 4633	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		nfv 1574	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐴
4		eleq1 2292	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
5	3, 4	ceqsalg 2828	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴) ↔ 𝐴 ∈ 𝐴))
6	2, 5	mtbiri 679	. . 3 ⊢ (𝐴 ∈ V → ¬ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴))
7		velsn 3683	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8		olc 716	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
9		elun 3345	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
10		ssid 3244	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
11		df-suc 4462	. . . . . . . . . . 11 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
12	11	eqeq1i 2237	. . . . . . . . . 10 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
13		sseq1 3247	. . . . . . . . . 10 ⊢ ((𝐴 ∪ {𝐴}) = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
14	12, 13	sylbi 121	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
15	10, 14	mpbiri 168	. . . . . . . 8 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∪ {𝐴}) ⊆ 𝐴)
16	15	sseld 3223	. . . . . . 7 ⊢ (suc 𝐴 = 𝐴 → (𝑥 ∈ (𝐴 ∪ {𝐴}) → 𝑥 ∈ 𝐴))
17	9, 16	biimtrrid 153	. . . . . 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	biimtrrid 153	. . . 4 ⊢ (suc 𝐴 = 𝐴 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐴))
20	19	alrimiv 1920	. . 3 ⊢ (suc 𝐴 = 𝐴 → ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴))
21	6, 20	nsyl3 629	. 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
22	1, 21	impbii 126	1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ wo 713  ∀wal 1393   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ∪ cun 3195   ⊆ wss 3197  {csn 3666  suc csuc 4456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211  ax-setind 4629

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-sn 3672  df-suc 4462

This theorem is referenced by: (None)