Theorem termcbasmo 50210

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 2774	. 2 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
2		eqeq2 2782	. 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 50207	. . . 4 ⊢ (𝜑 → ∃𝑧 𝐵 = {𝑧})
6		mosn 49540	. . . . 5 ⊢ (𝐵 = {𝑧} → ∃*𝑥 𝑥 ∈ 𝐵)
7	6	exlimiv 1958	. . . 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 3604	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Syntax hints:   → wi 4   = wceq 1568  ∃wex 1807   ∈ wcel 2150  ∃*wmo 2572  ∀wral 3086  {csn 4594  ‘cfv 6540  Basecbs 17272  TermCatctermc 50199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2152  ax-9 2160  ax-10 2183  ax-11 2199  ax-12 2220  ax-ext 2742

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2099  df-mo 2574  df-eu 2604  df-clab 2749  df-cleq 2762  df-clel 2845  df-nfc 2919  df-ral 3087  df-rex 3097  df-rmo 3376  df-reu 3377  df-rab 3424  df-v 3464  df-sbc 3753  df-dif 3916  df-un 3918  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-iota 6496  df-fv 6548  df-termc 50200

This theorem is referenced by:  termchomn0  50211  termchommo  50212  termcid  50213  termcid2  50214  termchom2  50216  termcarweu  50255  termfucterm  50271  cofuterm  50272  uobeqterm  50273