Theorem dfsuccf2 36705

Description: Alternate definition of Scott Fenton's version of Succ, cf. df-sucmap 39394. (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 36634	. 2 ⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
2		df-co 5660	. 2 ⊢ (Cup ∘ ( I ⊗ Singleton)) = {⟨𝑚, 𝑛⟩ ∣ ∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛)}
3		vex 3455	. . . . 5 ⊢ 𝑚 ∈ V
4		vex 3455	. . . . 5 ⊢ 𝑛 ∈ V
5	3, 4	lemsuccf 36703	. . . 4 ⊢ (∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛) ↔ 𝑛 = suc 𝑚)
6		eqcom 2768	. . . 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 5172	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ ∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛)} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
9	1, 2, 8	3eqtri 2788	1 ⊢ Succ = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}

This proof depends on syntax axioms:   ∧ wa 401   = wceq 1570  ∃wex 1812   class class class wbr 5103  {copab 5167   I cid 5545   ∘ ccom 5655  suc csuc 6364   ⊗ ctxp 36592  Singletoncsingle 36600  Cupccup 36608  Succcsuccf 36610

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 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391  ax-un 7751

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-symdif 4199  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-fo 6544  df-fv 6546  df-1st 8001  df-2nd 8002  df-txp 36616  df-singleton 36624  df-cup 36631  df-succf 36634

This theorem is used by: (None)