Theorem istermc2 49720

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 49719	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
3		eusn 4687	. . 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 1541  ∃wex 1780   ∈ wcel 2113  ∃!weu 2568  {csn 4580  ‘cfv 6492  Basecbs 17136  ThinCatcthinc 49662  TermCatctermc 49717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-uni 4864  df-br 5099  df-iota 6448  df-fv 6500  df-termc 49718

This theorem is referenced by:  arweutermc  49775  diag1f1o  49779  diag2f1o  49782