Theorem termchomn0 49489

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 2729	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		termcbas.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ TermCat)
5	4	termccd 49484	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		termcbasmo.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7	1, 2, 3, 5, 6	catidcl 17607	. . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
8		termcbasmo.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	4, 1, 6, 8	termcbasmo 49488	. . . 4 ⊢ (𝜑 → 𝑋 = 𝑌)
10	9	oveq2d 7369	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑋) = (𝑋𝐻𝑌))
11	7, 10	eleqtrd 2830	. 2 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑌))
12		n0i 4293	. 2 ⊢ (((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
13	11, 12	syl 17	1 ⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109  ∅c0 4286  ‘cfv 6486  (class class class)co 7353  Basecbs 17139  Hom chom 17191  Idccid 17590  TermCatctermc 49477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  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 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3345  df-reu 3346  df-rab 3397  df-v 3440  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-iun 4946  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-riota 7310  df-ov 7356  df-cat 17593  df-cid 17594  df-thinc 49423  df-termc 49478

This theorem is referenced by:  termchom  49493  functermc  49513  fulltermc  49516