Theorem termcbas 49179

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 49173	. . 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 1539  ∃wex 1778   ∈ wcel 2107  {csn 4606  ‘cfv 6541  Basecbs 17230  ThinCatcthinc 49118  TermCatctermc 49171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-iota 6494  df-fv 6549  df-termc 49172

This theorem is referenced by:  termcbas2  49180  termcbasmo  49181  oppctermhom  49202  functermc  49206  termcarweu  49226  diag1f1o  49232  diag2f1o  49235  basrestermcfo  49267