Theorem dfsuccl4 38497

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 38496	. 2 ⊢ Suc = {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛}
2		sucidg 6389	. . . . . . . . . 10 ⊢ (𝑚 ∈ V → 𝑚 ∈ suc 𝑚)
3	2	elv 3441	. . . . . . . . 9 ⊢ 𝑚 ∈ suc 𝑚
4		eleq2 2820	. . . . . . . . 9 ⊢ (suc 𝑚 = 𝑛 → (𝑚 ∈ suc 𝑚 ↔ 𝑚 ∈ 𝑛))
5	3, 4	mpbii 233	. . . . . . . 8 ⊢ (suc 𝑚 = 𝑛 → 𝑚 ∈ 𝑛)
6		sssucid 6388	. . . . . . . . 9 ⊢ 𝑚 ⊆ suc 𝑚
7		sseq2 3956	. . . . . . . . 9 ⊢ (suc 𝑚 = 𝑛 → (𝑚 ⊆ suc 𝑚 ↔ 𝑚 ⊆ 𝑛))
8	6, 7	mpbii 233	. . . . . . . 8 ⊢ (suc 𝑚 = 𝑛 → 𝑚 ⊆ 𝑛)
9	5, 8	jca 511	. . . . . . 7 ⊢ (suc 𝑚 = 𝑛 → (𝑚 ∈ 𝑛 ∧ 𝑚 ⊆ 𝑛))
10	9	pm4.71ri 560	. . . . . 6 ⊢ (suc 𝑚 = 𝑛 ↔ ((𝑚 ∈ 𝑛 ∧ 𝑚 ⊆ 𝑛) ∧ suc 𝑚 = 𝑛))
11		df-3an 1088	. . . . . 6 ⊢ ((𝑚 ∈ 𝑛 ∧ 𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛) ↔ ((𝑚 ∈ 𝑛 ∧ 𝑚 ⊆ 𝑛) ∧ suc 𝑚 = 𝑛))
12		3anass 1094	. . . . . 6 ⊢ ((𝑚 ∈ 𝑛 ∧ 𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛) ↔ (𝑚 ∈ 𝑛 ∧ (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)))
13	10, 11, 12	3bitr2i 299	. . . . 5 ⊢ (suc 𝑚 = 𝑛 ↔ (𝑚 ∈ 𝑛 ∧ (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)))
14	13	eubii 2580	. . . 4 ⊢ (∃!𝑚 suc 𝑚 = 𝑛 ↔ ∃!𝑚(𝑚 ∈ 𝑛 ∧ (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)))
15		df-reu 3347	. . . 4 ⊢ (∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛) ↔ ∃!𝑚(𝑚 ∈ 𝑛 ∧ (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)))
16	14, 15	bitr4i 278	. . 3 ⊢ (∃!𝑚 suc 𝑚 = 𝑛 ↔ ∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛))
17	16	abbii 2798	. 2 ⊢ {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛} = {𝑛 ∣ ∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)}
18	1, 17	eqtri 2754	1 ⊢ Suc = {𝑛 ∣ ∃!𝑚 ∈ 𝑛 (𝑚 ⊆ 𝑛 ∧ suc 𝑚 = 𝑛)}

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111  ∃!weu 2563  {cab 2709  ∃!wreu 3344  Vcvv 3436   ⊆ wss 3897  suc csuc 6308   Suc csuccl 38228

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668  ax-reg 9478

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-eprel 5514  df-fr 5567  df-cnv 5622  df-dm 5624  df-rn 5625  df-suc 6312  df-sucmap 38485  df-succl 38492

This theorem is referenced by: (None)