Theorem termchomn0 50262

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 2763	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		termcbas.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ TermCat)
5	4	termccd 50257	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		termcbasmo.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7	1, 2, 3, 5, 6	catidcl 17733	. . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
8		termcbasmo.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	4, 1, 6, 8	termcbasmo 50261	. . . 4 ⊢ (𝜑 → 𝑋 = 𝑌)
10	9	oveq2d 7426	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑋) = (𝑋𝐻𝑌))
11	7, 10	eleqtrd 2865	. 2 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑌))
12		n0i 4293	. 2 ⊢ (((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
13	11, 12	syl 18	1 ⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143  ∅c0 4286  ‘cfv 6536  (class class class)co 7410  Basecbs 17264  Hom chom 17316  Idccid 17716  TermCatctermc 50250

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-rep 5238  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-iun 4958  df-br 5110  df-opab 5174  df-mpt 5193  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-riota 7367  df-ov 7413  df-cat 17719  df-cid 17720  df-thinc 50196  df-termc 50251

This theorem is referenced by:  termchom  50266  functermc  50286  fulltermc  50289