Theorem termchomn0 49959

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 49954	. . . 4 ⊢ (𝜑 → 𝐶 ∈ Cat)
6		termcbasmo.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
7	1, 2, 3, 5, 6	catidcl 17648	. . 3 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑋))
8		termcbasmo.y	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
9	4, 1, 6, 8	termcbasmo 49958	. . . 4 ⊢ (𝜑 → 𝑋 = 𝑌)
10	9	oveq2d 7383	. . 3 ⊢ (𝜑 → (𝑋𝐻𝑋) = (𝑋𝐻𝑌))
11	7, 10	eleqtrd 2838	. 2 ⊢ (𝜑 → ((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑌))
12		n0i 4280	. 2 ⊢ (((Id‘𝐶)‘𝑋) ∈ (𝑋𝐻𝑌) → ¬ (𝑋𝐻𝑌) = ∅)
13	11, 12	syl 17	1 ⊢ (𝜑 → ¬ (𝑋𝐻𝑌) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1542   ∈ wcel 2114  ∅c0 4273  ‘cfv 6498  (class class class)co 7367  Basecbs 17179  Hom chom 17231  Idccid 17631  TermCatctermc 49947

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 2708  ax-rep 5212  ax-sep 5231  ax-nul 5241  ax-pr 5375

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 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3062  df-rmo 3342  df-reu 3343  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-iun 4935  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6454  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-fv 6506  df-riota 7324  df-ov 7370  df-cat 17634  df-cid 17635  df-thinc 49893  df-termc 49948

This theorem is referenced by:  termchom  49963  functermc  49983  fulltermc  49986