Theorem dfsuccl3 38724

Description: Alternate definition of the class of all successors. (Contributed by Peter Mazsa, 30-Jan-2026.)

Assertion

Ref	Expression
dfsuccl3	⊢ Suc = {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛}

Distinct variable group:   𝑚,𝑛

Proof of Theorem dfsuccl3

Step	Hyp	Ref	Expression
1		dfsuccl2 38721	. 2 ⊢ Suc = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}
2		exeupre2 38723	. . 3 ⊢ (∃𝑚 suc 𝑚 = 𝑛 ↔ ∃!𝑚 suc 𝑚 = 𝑛)
3	2	abbii 2804	. 2 ⊢ {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛} = {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛}
4	1, 3	eqtri 2760	1 ⊢ Suc = {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛}

Syntax hints:   = wceq 1542  ∃wex 1781  ∃!weu 2569  {cab 2715  suc csuc 6327   Suc csuccl 38430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-pr 5379  ax-un 7690  ax-reg 9509

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-eprel 5532  df-fr 5585  df-cnv 5640  df-dm 5642  df-rn 5643  df-suc 6331  df-sucmap 38713  df-succl 38720

This theorem is referenced by:  dfsuccl4  38725