Theorem istermc3 50942

Description: The predicate "is a terminal category". A terminal category is a thin category whose base set is equinumerous to 1_o. Consider en1b 9090, map1 9106, and euen1b 9093. (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 50940	. 2 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
3		en1 9089	. . 3 ⊢ (𝐵 ≈ 1o ↔ ∃𝑥 𝐵 = {𝑥})
4	3	anbi2i 635	. 2 ⊢ ((𝐶 ∈ ThinCat ∧ 𝐵 ≈ 1o) ↔ (𝐶 ∈ ThinCat ∧ ∃𝑥 𝐵 = {𝑥}))
5	2, 4	bitr4i 281	1 ⊢ (𝐶 ∈ TermCat ↔ (𝐶 ∈ ThinCat ∧ 𝐵 ≈ 1o))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {csn 4588   class class class wbr 5108  ‘cfv 6566  1_oc1o 8507   ≈ cen 9008  Basecbs 17431  ThinCatcthinc 50883  TermCatctermc 50938

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-12 2216  ax-ext 2735  ax-sep 5255  ax-nul 5266  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4841  df-uni 4873  df-br 5109  df-opab 5174  df-id 5553  df-xp 5664  df-rel 5665  df-cnv 5666  df-co 5667  df-dm 5668  df-rn 5669  df-res 5670  df-ima 5671  df-suc 6394  df-iota 6520  df-fun 6568  df-fn 6569  df-f 6570  df-f1 6571  df-fo 6572  df-f1o 6573  df-fv 6574  df-1o 8514  df-en 9012  df-termc 50939

This theorem is used by:  setcsnterm  50956  termcterm2  50980