Theorem termchomn0 49645

Description: All hom-sets of a terminal category are non-empty. (Contributed by Zhi Wang, 17-Oct-2025.)

Hypotheses

Ref	Expression
termcbas.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcbas.b	⊢ 𝐵 = (Base‘𝐶)
termcbasmo.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
termcbasmo.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)
termcid.h	⊢ 𝐻 = (Hom ‘𝐶)

Assertion

Ref	Expression
termchomn0	⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)

Proof of Theorem termchomn0

Step	Hyp	Ref	Expression
1		termcbas.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
2		termcid.h	. . . 4 ⊢ 𝐻 = (Hom ‘𝐶)
3		eqid 2733	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		termcbas.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ TermCat)
5	4	termccd 49640	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		termcbasmo.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7	1, 2, 3, 5, 6	catidcl 17596	. . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
8		termcbasmo.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	4, 1, 6, 8	termcbasmo 49644	. . . 4 ⊢ (𝜑 → 𝑋 = 𝑌)
10	9	oveq2d 7371	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑋) = (𝑋𝐻𝑌))
11	7, 10	eleqtrd 2835	. 2 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑌))
12		n0i 4289	. 2 ⊢ (((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
13	11, 12	syl 17	1 ⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1541   ∈ wcel 2113  ∅c0 4282  ‘cfv 6489  (class class class)co 7355  Basecbs 17127  Hom chom 17179  Idccid 17579  TermCatctermc 49633

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 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pr 5374

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 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-riota 7312  df-ov 7358  df-cat 17582  df-cid 17583  df-thinc 49579  df-termc 49634

This theorem is referenced by:  termchom  49649  functermc  49669  fulltermc  49672