Theorem dfsuccl4 39406

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 39405	. 2 ⊢ Suc = {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛}
2		sucidg 6446	. . . . . . . . . 10 ⊢ (𝑚 ∈ V → 𝑚 ∈ suc 𝑚)
3	2	elv 3456	. . . . . . . . 9 ⊢ 𝑚 ∈ suc 𝑚
4		eleq2 2850	. . . . . . . . 9 ⊢ (suc 𝑚 = 𝑛 → (𝑚 ∈ suc 𝑚 ↔ 𝑚 ∈ 𝑛))
5	3, 4	mpbii 236	. . . . . . . 8 ⊢ (suc 𝑚 = 𝑛 → 𝑚 ∈ 𝑛)
6		sssucid 6445	. . . . . . . . 9 ⊢ 𝑚 ⊆ suc 𝑚
7		sseq2 3957	. . . . . . . . 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 2611	. . . 4 ⊢ (∃!𝑚 suc 𝑚 = 𝑛 ↔ ∃!𝑚(𝑚 ∈ 𝑛 ∧ (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)))
15		df-reu 3367	. . . 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 2828	. 2 ⊢ {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛} = {𝑛 ∣ ∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)}
18	1, 17	eqtri 2784	1 ⊢ Suc = {𝑛 ∣ ∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)}

This proof depends on syntax axioms:   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∃!weu 2594  {cab 2739  ∃!wreu 3364  Vcvv 3451   ⊆ wss 3899  suc csuc 6364   Suc csuccl 39111

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-pr 5391  ax-un 7751  ax-reg 9586

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-reu 3367  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-eprel 5551  df-fr 5604  df-cnv 5659  df-dm 5661  df-rn 5662  df-suc 6368  df-sucmap 39394  df-succl 39401

This theorem is used by: (None)