Theorem termchomn0 49729

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 2736	. . . 4 ⊢ (Id‘𝐶) = (Id‘𝐶)
4		termcbas.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ TermCat)
5	4	termccd 49724	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		termcbasmo.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7	1, 2, 3, 5, 6	catidcl 17605	. . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
8		termcbasmo.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	4, 1, 6, 8	termcbasmo 49728	. . . 4 ⊢ (𝜑 → 𝑋 = 𝑌)
10	9	oveq2d 7374	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑋) = (𝑋𝐻𝑌))
11	7, 10	eleqtrd 2838	. 2 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑌))
12		n0i 4292	. 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 4285  ‘cfv 6492  (class class class)co 7358  Basecbs 17136  Hom chom 17188  Idccid 17588  TermCatctermc 49717

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 2184  ax-ext 2708  ax-rep 5224  ax-sep 5241  ax-nul 5251  ax-pr 5377

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rmo 3350  df-reu 3351  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-iun 4948  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-iota 6448  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-fv 6500  df-riota 7315  df-ov 7361  df-cat 17591  df-cid 17592  df-thinc 49663  df-termc 49718

This theorem is referenced by:  termchom  49733  functermc  49753  fulltermc  49756