Theorem termcbas 49485

Description: The base of a terminal category is a singleton. (Contributed by Zhi Wang, 16-Oct-2025.)

Hypotheses

Ref	Expression
termcbas.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcbas.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
termcbas	⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})

Distinct variable group:   𝑥,𝐶

Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)

Proof of Theorem termcbas

Step	Hyp	Ref	Expression
1		termcbas.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termcbas.b	. . . 4 ⊢ 𝐵 = (Base‘𝐶)
3	2	istermc 49479	. . 3 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
4	1, 3	sylib 218	. 2 ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
5	4	simprd 495	1 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {csn 4579  ‘cfv 6486  Basecbs 17139  ThinCatcthinc 49422  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-ext 2701

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-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-iota 6442  df-fv 6494  df-termc 49478

This theorem is referenced by:  termco  49486  termcbas2  49487  termcbasmo  49488  oppctermhom  49509  functermc  49513  termcarweu  49533  diag1f1o  49539  diag2f1o  49542  basrestermcfo  49580