Theorem istermc3 50058

Description: The predicate "is a terminal category". A terminal category is a thin category whose base set is equinumerous to 1_o. Consider en1b 9000, map1 9015, and euen1b 9003. (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 50056	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
3		en1 8999	. . 3 ⊢ (𝐵 ≈ 1o ↔ ∃𝑥 𝐵 = {𝑥})
4	3	anbi2i 632	. 2 ⊢ ((𝐶 ∈ ThinCat ∧ 𝐵 ≈ 1o) ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
5	2, 4	bitr4i 280	1 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ 𝐵 ≈ 1o))

Syntax hints:   ↔ wb 208   ∧ wa 399   = wceq 1559  ∃wex 1798   ∈ wcel 2141  {csn 4579   class class class wbr 5097  ‘cfv 6516  1_oc1o 8424   ≈ cen 8918  Basecbs 17236  ThinCatcthinc 49999  TermCatctermc 50054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3076  df-rex 3086  df-reu 3367  df-rab 3414  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-suc 6347  df-iota 6472  df-fun 6518  df-fn 6519  df-f 6520  df-f1 6521  df-fo 6522  df-f1o 6523  df-fv 6524  df-1o 8431  df-en 8922  df-termc 50055

This theorem is referenced by:  setcsnterm  50072  termcterm2  50096