Theorem sucprcreg 4566

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 4430	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 4558	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		nfv 1539	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐴
4		eleq1 2252	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
5	3, 4	ceqsalg 2780	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴) ↔ 𝐴 ∈ 𝐴))
6	2, 5	mtbiri 676	. . 3 ⊢ (𝐴 ∈ V → ¬ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴))
7		velsn 3624	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8		olc 712	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
9		elun 3291	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
10		ssid 3190	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
11		df-suc 4389	. . . . . . . . . . 11 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
12	11	eqeq1i 2197	. . . . . . . . . 10 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
13		sseq1 3193	. . . . . . . . . 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 3169	. . . . . . 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 1885	. . 3 ⊢ (suc 𝐴 = 𝐴 → ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴))
21	6, 20	nsyl3 627	. 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 709  ∀wal 1362   = wceq 1364   ∈ wcel 2160  Vcvv 2752   ∪ cun 3142   ⊆ wss 3144  {csn 3607  suc csuc 4383

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171  ax-setind 4554

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-v 2754  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-sn 3613  df-suc 4389

This theorem is referenced by: (None)