Theorem termcbasmo 49181

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 2738	. 2 ⊢ (𝑥 = 𝑋 → (𝑥 = 𝑦 ↔ 𝑋 = 𝑦))
2		eqeq2 2746	. 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 49179	. . . 4 ⊢ (𝜑 → ∃𝑧 𝐵 = {𝑧})
6		mosn 48705	. . . . 5 ⊢ (𝐵 = {𝑧} → ∃*𝑥 𝑥 ∈ 𝐵)
7	6	exlimiv 1929	. . . 4 ⊢ (∃𝑧 𝐵 = {𝑧} → ∃*𝑥 𝑥 ∈ 𝐵)
8	5, 7	syl 17	. . 3 ⊢ (𝜑 → ∃*𝑥 𝑥 ∈ 𝐵)
9		moel 3385	. . 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 3620	1 ⊢ (𝜑 → 𝑋 = 𝑌)

Syntax hints:   → wi 4   = wceq 1539  ∃wex 1778   ∈ wcel 2107  ∃*wmo 2536  ∀wral 3050  {csn 4606  ‘cfv 6541  Basecbs 17230  TermCatctermc 49171

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-nfc 2884  df-ral 3051  df-rex 3060  df-rmo 3363  df-reu 3364  df-rab 3420  df-v 3465  df-sbc 3771  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-iota 6494  df-fv 6549  df-termc 49172

This theorem is referenced by:  termchomn0  49182  termchommo  49183  termcid  49184  termcid2  49185  termchom2  49187  termcarweu  49226