Theorem termcbasmo 49644

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 2737	. 2 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
2		eqeq2 2745	. 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 49641	. . . 4 ⊢ (𝜑 → ∃𝑧 𝐵 = {𝑧})
6		mosn 48974	. . . . 5 ⊢ (𝐵 = {𝑧} → ∃*𝑥 𝑥 ∈ 𝐵)
7	6	exlimiv 1931	. . . 4 ⊢ (∃𝑧 𝐵 = {𝑧} → ∃*𝑥 𝑥 ∈ 𝐵)
8	5, 7	syl 17	. . 3 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)
9		moel 3367	. . 3 ⊢ (∃*𝑥 𝑥 ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 𝑥 = 𝑦)
10	8, 9	sylib 218	. 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 3588	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Syntax hints:   → wi 4   = wceq 1541  ∃wex 1780   ∈ wcel 2113  ∃*wmo 2535  ∀wral 3048  {csn 4577  ‘cfv 6489  Basecbs 17127  TermCatctermc 49633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ral 3049  df-rex 3058  df-rmo 3347  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-br 5096  df-iota 6445  df-fv 6497  df-termc 49634

This theorem is referenced by:  termchomn0  49645  termchommo  49646  termcid  49647  termcid2  49648  termchom2  49650  termcarweu  49689  termfucterm  49705  cofuterm  49706  uobeqterm  49707