Theorem sucprcreg 4691

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 4552	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 4683	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		nfv 1581	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐴
4		eleq1 2301	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
5	3, 4	ceqsalg 2850	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴) ↔ 𝐴 ∈ 𝐴))
6	2, 5	mtbiri 686	. . 3 ⊢ (𝐴 ∈ V → ¬ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴))
7		velsn 3722	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8		olc 723	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
9		elun 3370	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
10		ssid 3268	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
11		df-suc 4511	. . . . . . . . . . 11 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
12	11	eqeq1i 2246	. . . . . . . . . 10 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
13		sseq1 3271	. . . . . . . . . 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 3247	. . . . . . 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 1927	. . 3 ⊢ (suc 𝐴 = 𝐴 → ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴))
21	6, 20	nsyl3 635	. 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 720  ∀wal 1400   = wceq 1402   ∈ wcel 2209  Vcvv 2821   ∪ cun 3218   ⊆ wss 3220  {csn 3705  suc csuc 4505

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220  ax-setind 4679

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-sn 3711  df-suc 4511

This theorem is referenced by: (None)