Theorem termcbas2 50094

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 50092	. 2 ⊢ (𝜑 → ∃𝑥 𝐵 = {𝑥})
4		simpr 488	. . 3 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝐵 = {𝑥})
5		termcbasmo.x	. . . . . 6 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
6	5	adantr 484	. . . . 5 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝑋 ∈ 𝐵)
7	6, 4	eleqtrd 2865	. . . 4 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝑋 ∈ {𝑥})
8		elsni 4600	. . . . 5 ⊢ (𝑋 ∈ {𝑥} → 𝑋 = 𝑥)
9	8	sneqd 4595	. . . 4 ⊢ (𝑋 ∈ {𝑥} → {𝑋} = {𝑥})
10	7, 9	syl 17	. . 3 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → {𝑋} = {𝑥})
11	4, 10	eqtr4d 2801	. 2 ⊢ ((𝜑 ∧ 𝐵 = {𝑥}) → 𝐵 = {𝑋})
12	3, 11	exlimddv 1956	1 ⊢ (𝜑 → 𝐵 = {𝑋})

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1561   ∈ wcel 2143  {csn 4583  ‘cfv 6521  Basecbs 17255  TermCatctermc 50084

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-clab 2742  df-cleq 2755  df-clel 2838  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5102  df-iota 6477  df-fv 6529  df-termc 50085

This theorem is referenced by:  termcfuncval  50144  diag1f1olem  50145  termcnatval  50147  diag2f1olem  50148