Theorem termcbas2 50822

Description: The base of a terminal category is given by its object. (Contributed by Zhi Wang, 20-Oct-2025.)

Hypotheses

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

Assertion

Ref	Expression
termcbas2	⊢ (𝜑 → 𝐵 = {𝑋})

Proof of Theorem termcbas2

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		termcbas.c	. . 3 ⊢ (𝜑 → 𝐶 ∈ TermCat)
2		termcbas.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
3	1, 2	termcbas 50820	. 2 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})
4		simpr 490	. . 3 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝐵 = {𝑥})
5		termcbasmo.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6	5	adantr 486	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝑋 ∈ 𝐵)
7	6, 4	eleqtrd 2866	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝑋 ∈ {𝑥})
8		elsni 4601	. . . . 5 ⊢ (𝑋 ∈ {𝑥} → 𝑋 = 𝑥)
9	8	sneqd 4596	. . . 4 ⊢ (𝑋 ∈ {𝑥} → {𝑋} = {𝑥})
10	7, 9	syl 18	. . 3 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → {𝑋} = {𝑥})
11	4, 10	eqtr4d 2801	. 2 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝐵 = {𝑋})
12	3, 11	exlimddv 1970	1 ⊢ (𝜑 → 𝐵 = {𝑋})

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570   ∈ wcel 2147  {csn 4588  ‘cfv 6558  Basecbs 17416  TermCatctermc 50812

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-iota 6512  df-fv 6566  df-termc 50813

This theorem is used by:  termcfuncval  50872  diag1f1olem  50873  termcnatval  50878  diag2f1olem  50879