Theorem dfsuccf2 36331

Description: Alternate definition of Scott Fenton's version of Succ, cf. df-sucmap 39000. (Contributed by Peter Mazsa, 6-Jan-2026.)

Assertion

Ref	Expression
dfsuccf2	⊢ Succ = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}

Distinct variable group:   𝑚,𝑛

Proof of Theorem dfsuccf2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-succf 36260	. 2 ⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
2		df-co 5671	. 2 ⊢ (Cup ∘ ( I ⊗ Singleton)) = {⟨𝑚, 𝑛⟩ ∣ ∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛)}
3		vex 3467	. . . . 5 ⊢ 𝑚 ∈ V
4		vex 3467	. . . . 5 ⊢ 𝑛 ∈ V
5	3, 4	lemsuccf 36329	. . . 4 ⊢ (∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛) ↔ 𝑛 = suc 𝑚)
6		eqcom 2776	. . . 4 ⊢ (𝑛 = suc 𝑚 ↔ suc 𝑚 = 𝑛)
7	5, 6	bitri 278	. . 3 ⊢ (∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛) ↔ suc 𝑚 = 𝑛)
8	7	opabbii 5182	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ ∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛)} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
9	1, 2, 8	3eqtri 2796	1 ⊢ Succ = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}

Syntax hints:   ∧ wa 400   = wceq 1567  ∃wex 1806   class class class wbr 5113  {copab 5177   I cid 5556   ∘ ccom 5666  suc csuc 6363   ⊗ ctxp 36218  Singletoncsingle 36226  Cupccup 36234  Succcsuccf 36236

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405  ax-un 7733

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-symdif 4214  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4877  df-br 5114  df-opab 5178  df-mpt 5197  df-id 5557  df-eprel 5562  df-xp 5668  df-rel 5669  df-cnv 5670  df-co 5671  df-dm 5672  df-rn 5673  df-res 5674  df-suc 6367  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fo 6543  df-fv 6545  df-1st 7985  df-2nd 7986  df-txp 36242  df-singleton 36250  df-cup 36257  df-succf 36260

This theorem is referenced by: (None)