Theorem dfsuccf2 36470

Description: Alternate definition of Scott Fenton's version of Succ, cf. df-sucmap 39169. (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 36399	. 2 ⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
2		df-co 5672	. 2 ⊢ (Cup ∘ ( I ⊗ Singleton)) = {⟨𝑚, 𝑛⟩ ∣ ∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛)}
3		vex 3461	. . . . 5 ⊢ 𝑚 ∈ V
4		vex 3461	. . . . 5 ⊢ 𝑛 ∈ V
5	3, 4	lemsuccf 36468	. . . 4 ⊢ (∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛) ↔ 𝑛 = suc 𝑚)
6		eqcom 2772	. . . 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 5180	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ ∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛)} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
9	1, 2, 8	3eqtri 2792	1 ⊢ Succ = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   class class class wbr 5111  {copab 5175   I cid 5557   ∘ ccom 5667  suc csuc 6366   ⊗ ctxp 36357  Singletoncsingle 36365  Cupccup 36373  Succcsuccf 36375

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-nul 5271  ax-pr 5406  ax-un 7742

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 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-symdif 4206  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4875  df-br 5112  df-opab 5176  df-mpt 5195  df-id 5558  df-eprel 5563  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-suc 6370  df-iota 6496  df-fun 6542  df-fn 6543  df-f 6544  df-fo 6546  df-fv 6548  df-1st 7992  df-2nd 7993  df-txp 36381  df-singleton 36389  df-cup 36396  df-succf 36399

This theorem is used by: (None)