Theorem istermc3 49445

Description: The predicate "is a terminal category". A terminal category is a thin category whose base set is equinumerous to 1_o. Consider en1b 8998, map1 9013, and euen1b 9001. (Contributed by Zhi Wang, 16-Oct-2025.)

Hypothesis

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

Assertion

Ref	Expression
istermc3	⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ 𝐵 ≈ 1o))

Proof of Theorem istermc3

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		istermc.b	. . 3 ⊢ 𝐵 = (Base‘𝐶)
2	1	istermc 49443	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
3		en1 8997	. . 3 ⊢ (𝐵 ≈ 1o ↔ ∃𝑥 𝐵 = {𝑥})
4	3	anbi2i 623	. 2 ⊢ ((𝐶 ∈ ThinCat ∧ 𝐵 ≈ 1o) ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
5	2, 4	bitr4i 278	1 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ 𝐵 ≈ 1o))

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {csn 4591   class class class wbr 5109  ‘cfv 6513  1_oc1o 8429   ≈ cen 8917  Basecbs 17185  ThinCatcthinc 49386  TermCatctermc 49441

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-id 5535  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-ima 5653  df-suc 6340  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-1o 8436  df-en 8921  df-termc 49442

This theorem is referenced by:  setcsnterm  49459  termcterm2  49483