Theorem sucprcreg 4273

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 4149	. 2 ⊢ (¬ 𝐴 ∈ V → suc 𝐴 = 𝐴)
2		elirr 4266	. . . 4 ⊢ ¬ 𝐴 ∈ 𝐴
3		nfv 1421	. . . . 5 ⊢ Ⅎ𝑥 𝐴 ∈ 𝐴
4		eleq1 2100	. . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝐴 ↔ 𝐴 ∈ 𝐴))
5	3, 4	ceqsalg 2582	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴) ↔ 𝐴 ∈ 𝐴))
6	2, 5	mtbiri 600	. . 3 ⊢ (𝐴 ∈ V → ¬ ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴))
7		velsn 3392	. . . . 5 ⊢ (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴)
8		olc 632	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
9		elun 3084	. . . . . . 7 ⊢ (𝑥 ∈ (𝐴 ∪ {𝐴}) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}))
10		ssid 2964	. . . . . . . . 9 ⊢ 𝐴 ⊆ 𝐴
11		df-suc 4108	. . . . . . . . . . 11 ⊢ suc 𝐴 = (𝐴 ∪ {𝐴})
12	11	eqeq1i 2047	. . . . . . . . . 10 ⊢ (suc 𝐴 = 𝐴 ↔ (𝐴 ∪ {𝐴}) = 𝐴)
13		sseq1 2966	. . . . . . . . . 10 ⊢ ((𝐴 ∪ {𝐴}) = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
14	12, 13	sylbi 114	. . . . . . . . 9 ⊢ (suc 𝐴 = 𝐴 → ((𝐴 ∪ {𝐴}) ⊆ 𝐴 ↔ 𝐴 ⊆ 𝐴))
15	10, 14	mpbiri 157	. . . . . . . 8 ⊢ (suc 𝐴 = 𝐴 → (𝐴 ∪ {𝐴}) ⊆ 𝐴)
16	15	sseld 2944	. . . . . . 7 ⊢ (suc 𝐴 = 𝐴 → (𝑥 ∈ (𝐴 ∪ {𝐴}) → 𝑥 ∈ 𝐴))
17	9, 16	syl5bir 142	. . . . . 6 ⊢ (suc 𝐴 = 𝐴 → ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ {𝐴}) → 𝑥 ∈ 𝐴))
18	8, 17	syl5 28	. . . . 5 ⊢ (suc 𝐴 = 𝐴 → (𝑥 ∈ {𝐴} → 𝑥 ∈ 𝐴))
19	7, 18	syl5bir 142	. . . 4 ⊢ (suc 𝐴 = 𝐴 → (𝑥 = 𝐴 → 𝑥 ∈ 𝐴))
20	19	alrimiv 1754	. . 3 ⊢ (suc 𝐴 = 𝐴 → ∀𝑥(𝑥 = 𝐴 → 𝑥 ∈ 𝐴))
21	6, 20	nsyl3 556	. 2 ⊢ (suc 𝐴 = 𝐴 → ¬ 𝐴 ∈ V)
22	1, 21	impbii 117	1 ⊢ (¬ 𝐴 ∈ V ↔ suc 𝐴 = 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ wo 629  ∀wal 1241   = wceq 1243   ∈ wcel 1393  Vcvv 2557   ∪ cun 2915   ⊆ wss 2917  {csn 3375  suc csuc 4102

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-setind 4262

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-ral 2311  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-sn 3381  df-suc 4108

This theorem is referenced by: (None)