Theorem dfsuccl3 39671

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 39668	. 2 ⊢ Suc = {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛}
2		exeupre2 39670	. . 3 ⊢ (∃𝑚 suc 𝑚 = 𝑛 ↔ ∃!𝑚 suc 𝑚 = 𝑛)
3	2	abbii 2830	. 2 ⊢ {𝑛 ∣ ∃𝑚 suc 𝑚 = 𝑛} = {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛}
4	1, 3	eqtri 2786	1 ⊢ Suc = {𝑛 ∣ ∃!𝑚 suc 𝑚 = 𝑛}

This proof depends on syntax axioms:   = wceq 1570  ∃wex 1812  ∃!weu 2596  {cab 2741  suc csuc 6390   Suc csuccl 39377

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-pr 5398  ax-un 7783  ax-reg 9623

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 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-pw 4562  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-eprel 5558  df-fr 5611  df-cnv 5666  df-dm 5668  df-rn 5669  df-suc 6394  df-sucmap 39660  df-succl 39667

This theorem is used by:  dfsuccl4  39672