Theorem termcbasmo 49445

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 2733	. 2 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
2		eqeq2 2741	. 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 49442	. . . 4 ⊢ (𝜑 → ∃𝑧 𝐵 = {𝑧})
6		mosn 48774	. . . . 5 ⊢ (𝐵 = {𝑧} → ∃*𝑥 𝑥 ∈ 𝐵)
7	6	exlimiv 1930	. . . 4 ⊢ (∃𝑧 𝐵 = {𝑧} → ∃*𝑥 𝑥 ∈ 𝐵)
8	5, 7	syl 17	. . 3 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)
9		moel 3373	. . 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 3600	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Syntax hints:   → wi 4   = wceq 1540  ∃wex 1779   ∈ wcel 2109  ∃*wmo 2531  ∀wral 3044  {csn 4585  ‘cfv 6499  Basecbs 17155  TermCatctermc 49434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rmo 3351  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-dif 3914  df-un 3916  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-iota 6452  df-fv 6507  df-termc 49435

This theorem is referenced by:  termchomn0  49446  termchommo  49447  termcid  49448  termcid2  49449  termchom2  49451  termcarweu  49490  termfucterm  49506  cofuterm  49507  uobeqterm  49508