Theorem dfsuccl4 39183

Description: Alternate definition that incorporates the most desirable properties of the successor class. (Contributed by Peter Mazsa, 30-Jan-2026.)

Assertion

Ref	Expression
dfsuccl4	⊢ Suc = {𝑛 ∣ ∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)}

Distinct variable group:   𝑚,𝑛

Proof of Theorem dfsuccl4

Step	Hyp	Ref	Expression
1		dfsuccl3 39182	. 2 ⊢ Suc = {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛}
2		sucidg 6448	. . . . . . . . . 10 ⊢ (𝑚 ∈ V → 𝑚 ∈ suc 𝑚)
3	2	elv 3462	. . . . . . . . 9 ⊢ 𝑚 ∈ suc 𝑚
4		eleq2 2854	. . . . . . . . 9 ⊢ (suc 𝑚 = 𝑛 → (𝑚 ∈ suc 𝑚 ↔ 𝑚 ∈ 𝑛))
5	3, 4	mpbii 236	. . . . . . . 8 ⊢ (suc 𝑚 = 𝑛 → 𝑚 ∈ 𝑛)
6		sssucid 6447	. . . . . . . . 9 ⊢ 𝑚 ⊆ suc 𝑚
7		sseq2 3964	. . . . . . . . 9 ⊢ (suc 𝑚 = 𝑛 → (𝑚 ⊆ suc 𝑚 ↔ 𝑚 ⊆ 𝑛))
8	6, 7	mpbii 236	. . . . . . . 8 ⊢ (suc 𝑚 = 𝑛 → 𝑚 ⊆ 𝑛)
9	5, 8	jca 521	. . . . . . 7 ⊢ (suc 𝑚 = 𝑛 → (𝑚 ∈ 𝑛 ∧ 𝑚 ⊆ 𝑛))
10	9	pm4.71ri 570	. . . . . 6 ⊢ (suc 𝑚 = 𝑛 ↔ ((𝑚 ∈ 𝑛 ∧ 𝑚 ⊆ 𝑛) ∧ suc 𝑚 = 𝑛))
11		df-3an 1105	. . . . . 6 ⊢ ((𝑚 ∈ 𝑛 ∧ 𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛) ↔ ((𝑚 ∈ 𝑛 ∧ 𝑚 ⊆ 𝑛) ∧ suc 𝑚 = 𝑛))
12		3anass 1111	. . . . . 6 ⊢ ((𝑚 ∈ 𝑛 ∧ 𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛) ↔ (𝑚 ∈ 𝑛 ∧ (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)))
13	10, 11, 12	3bitr2i 302	. . . . 5 ⊢ (suc 𝑚 = 𝑛 ↔ (𝑚 ∈ 𝑛 ∧ (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)))
14	13	eubii 2615	. . . 4 ⊢ (∃!𝑚 suc 𝑚 = 𝑛 ↔ ∃!𝑚(𝑚 ∈ 𝑛 ∧ (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)))
15		df-reu 3372	. . . 4 ⊢ (∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛) ↔ ∃!𝑚(𝑚 ∈ 𝑛 ∧ (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)))
16	14, 15	bitr4i 281	. . 3 ⊢ (∃!𝑚 suc 𝑚 = 𝑛 ↔ ∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛))
17	16	abbii 2832	. 2 ⊢ {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛} = {𝑛 ∣ ∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)}
18	1, 17	eqtri 2788	1 ⊢ Suc = {𝑛 ∣ ∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)}

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2146  ∃!weu 2598  {cab 2743  ∃!wreu 3369  Vcvv 3457   ⊆ wss 3906  suc csuc 6366   Suc csuccl 38888

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406  ax-un 7742  ax-reg 9561

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-reu 3372  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-pw 4566  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-eprel 5563  df-fr 5616  df-cnv 5671  df-dm 5673  df-rn 5674  df-suc 6370  df-sucmap 39171  df-succl 39178

This theorem is used by: (None)