Theorem dfsuccf2 36084

Description: Alternate definition of Scott Fenton's version of Succ, cf. df-sucmap 38575. (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 36013	. 2 ⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
2		df-co 5631	. 2 ⊢ (Cup ∘ ( I ⊗ Singleton)) = {⟨𝑚, 𝑛⟩ ∣ ∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛)}
3		vex 3442	. . . . 5 ⊢ 𝑚 ∈ V
4		vex 3442	. . . . 5 ⊢ 𝑛 ∈ V
5	3, 4	lemsuccf 36082	. . . 4 ⊢ (∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛) ↔ 𝑛 = suc 𝑚)
6		eqcom 2741	. . . 4 ⊢ (𝑛 = suc 𝑚 ↔ suc 𝑚 = 𝑛)
7	5, 6	bitri 275	. . 3 ⊢ (∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛) ↔ suc 𝑚 = 𝑛)
8	7	opabbii 5163	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ ∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛)} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
9	1, 2, 8	3eqtri 2761	1 ⊢ Succ = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}

Syntax hints:   ∧ wa 395   = wceq 1541  ∃wex 1780   class class class wbr 5096  {copab 5158   I cid 5516   ∘ ccom 5626  suc csuc 6317   ⊗ ctxp 35971  Singletoncsingle 35979  Cupccup 35987  Succcsuccf 35989

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-symdif 4203  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-eprel 5522  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-fo 6496  df-fv 6498  df-1st 7931  df-2nd 7932  df-txp 35995  df-singleton 36003  df-cup 36010  df-succf 36013

This theorem is referenced by: (None)