Theorem dfsuccf2 36157

Description: Alternate definition of Scott Fenton's version of Succ, cf. df-sucmap 38713. (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 36086	. 2 ⊢ Succ = (Cup ∘ ( I ⊗ Singleton))
2		df-co 5641	. 2 ⊢ (Cup ∘ ( I ⊗ Singleton)) = {⟨𝑚, 𝑛⟩ ∣ ∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛)}
3		vex 3446	. . . . 5 ⊢ 𝑚 ∈ V
4		vex 3446	. . . . 5 ⊢ 𝑛 ∈ V
5	3, 4	lemsuccf 36155	. . . 4 ⊢ (∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛) ↔ 𝑛 = suc 𝑚)
6		eqcom 2744	. . . 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 5167	. 2 ⊢ {⟨𝑚, 𝑛⟩ ∣ ∃𝑥(𝑚( I ⊗ Singleton)𝑥 ∧ 𝑥Cup𝑛)} = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}
9	1, 2, 8	3eqtri 2764	1 ⊢ Succ = {⟨𝑚, 𝑛⟩ ∣ suc 𝑚 = 𝑛}

Syntax hints:   ∧ wa 395   = wceq 1542  ∃wex 1781   class class class wbr 5100  {copab 5162   I cid 5526   ∘ ccom 5636  suc csuc 6327   ⊗ ctxp 36044  Singletoncsingle 36052  Cupccup 36060  Succcsuccf 36062

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5243  ax-nul 5253  ax-pr 5379  ax-un 7690

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-symdif 4207  df-nul 4288  df-if 4482  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-mpt 5182  df-id 5527  df-eprel 5532  df-xp 5638  df-rel 5639  df-cnv 5640  df-co 5641  df-dm 5642  df-rn 5643  df-res 5644  df-suc 6331  df-iota 6456  df-fun 6502  df-fn 6503  df-f 6504  df-fo 6506  df-fv 6508  df-1st 7943  df-2nd 7944  df-txp 36068  df-singleton 36076  df-cup 36083  df-succf 36086

This theorem is referenced by: (None)