Theorem termcbas 50062

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 50056	. . 3 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
4	1, 3	sylib 220	. 2 ⊢ (𝜑 → (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
5	4	simprd 499	1 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {csn 4579  ‘cfv 6516  Basecbs 17236  ThinCatcthinc 49999  TermCatctermc 50054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-iota 6472  df-fv 6524  df-termc 50055

This theorem is referenced by:  termco  50063  termcbas2  50064  termcbasmo  50065  oppctermhom  50086  functermc  50090  termcarweu  50110  diag1f1o  50116  diag2f1o  50119  basrestermcfo  50157