Theorem termcbasmo 50141

Description: Two objects in a terminal category are identical. (Contributed by Zhi Wang, 16-Oct-2025.)

Hypotheses

Ref	Expression
termcbas.c	⊢ (𝜑 → 𝐶 ∈ TermCat)
termcbas.b	⊢ 𝐵 = (Base‘𝐶)
termcbasmo.x	⊢ (𝜑 → 𝑋 ∈ 𝐵)
termcbasmo.y	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
termcbasmo	⊢ (𝜑 → 𝑋 = 𝑌)

Proof of Theorem termcbasmo

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqeq1 2773	. 2 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
2		eqeq2 2781	. 2 ⊢ (𝑦 = 𝑌 → (𝑋 = 𝑦 ↔ 𝑋 = 𝑌))
3		termcbas.c	. . . . 5 ⊢ (𝜑 → 𝐶 ∈ TermCat)
4		termcbas.b	. . . . 5 ⊢ 𝐵 = (Base‘𝐶)
5	3, 4	termcbas 50138	. . . 4 ⊢ (𝜑 → ∃𝑧 𝐵 = {𝑧})
6		mosn 49471	. . . . 5 ⊢ (𝐵 = {𝑧} → ∃*𝑥 𝑥 ∈ 𝐵)
7	6	exlimiv 1957	. . . 4 ⊢ (∃𝑧 𝐵 = {𝑧} → ∃*𝑥 𝑥 ∈ 𝐵)
8	5, 7	syl 18	. . 3 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)
9		moel 3396	. . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝑥 = 𝑦)
10	8, 9	sylib 221	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝑥 = 𝑦)
11		termcbasmo.x	. 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵)
12		termcbasmo.y	. 2 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
13	1, 2, 10, 11, 12	rspc2dv 3605	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Syntax hints:   → wi 4   = wceq 1567  ∃wex 1806   ∈ wcel 2149  ∃*wmo 2571  ∀wral 3085  {csn 4591  ‘cfv 6534  Basecbs 17265  TermCatctermc 50130

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3465  df-sbc 3754  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5111  df-iota 6490  df-fv 6542  df-termc 50131

This theorem is referenced by:  termchomn0  50142  termchommo  50143  termcid  50144  termcid2  50145  termchom2  50147  termcarweu  50186  termfucterm  50202  cofuterm  50203  uobeqterm  50204