Theorem istermc2 49095

Description: The predicate "is a terminal category". A terminal category is a thin category with exactly one object. (Contributed by Zhi Wang, 16-Oct-2025.)

Hypothesis

Ref	Expression
istermc.b	⊢ 𝐵 = (Base‘𝐶)

Assertion

Ref	Expression
istermc2	⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃!𝑥 𝑥 ∈ 𝐵))

Distinct variable groups:   𝑥,𝐶   𝑥,𝐵

Proof of Theorem istermc2

Step	Hyp	Ref	Expression
1		istermc.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2	1	istermc 49094	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
3		eusn 4728	. . 3 ⊢ (∃!𝑥 𝑥 ∈ 𝐵 ↔ ∃𝑥 𝐵 = {𝑥})
4	3	anbi2i 623	. 2 ⊢ ((𝐶 ∈ ThinCat ∧ ∃!𝑥 𝑥 ∈ 𝐵) ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
5	2, 4	bitr4i 278	1 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃!𝑥 𝑥 ∈ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  ∃!weu 2567  {csn 4624  ‘cfv 6559  Basecbs 17243  ThinCatcthinc 49040  TermCatctermc 49092

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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-iota 6512  df-fv 6567  df-termc 49093

This theorem is referenced by: (None)