Theorem parhomeqor2 50963

Description: Variant of parhomeqor 50964. (Contributed by Zhi Wang, 14-Sep-2026.) (Revised by Zhi Wang, 17-Sep-2026.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypotheses

Ref	Expression
parhomval.h	⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
parhomeqor.f	⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
parhomeqor.g	⊢ (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))
parhomeqor2.s	⊢ 𝑆 ∈ V
parhomeqor2.j	⊢ 𝐽 ∈ V
parhomeqor2.n	⊢ 𝑋 ≠ 𝑌
parhomeqor2.u	⊢ (𝜑 → 𝑈 ∈ 𝐵)
parhomeqor2.v	⊢ (𝜑 → 𝑉 ∈ 𝐵)
parhomeqor2.w	⊢ (𝜑 → 𝑊 ∈ 𝐵)

Assertion

Ref	Expression
parhomeqor2	⊢ (𝜑 → (𝑈 = 𝑉 ∨ 𝑉 = 𝑊))

Distinct variable groups:   𝑥,𝐵,𝑦   𝑥,𝐽,𝑦   𝑥,𝑆,𝑦   𝑥,𝑈,𝑦   𝑥,𝑋,𝑦   𝑥,𝑌,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦

Allowed substitution hints:   𝜑(𝑥, 𝑦)   𝐹(𝑥, 𝑦)   𝐺(𝑥, 𝑦)   𝐻(𝑥, 𝑦)

Proof of Theorem parhomeqor2

Step	Hyp	Ref	Expression
1		parhomval.h	. . . . . . 7 ⊢ 𝐻 = (𝑥 ∈ 𝐵, 𝑦 ∈ 𝐵 ↦ if(𝑥 = 𝑦, {𝐽}, if((𝑥 = 𝑋 ∧ 𝑦 = 𝑌), 𝑆, ∅)))
2		parhomeqor.g	. . . . . . . 8 ⊢ (𝜑 → 𝐺 ∈ (𝑉𝐻𝑊))
3	2	ad2antrr 739	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → 𝐺 ∈ (𝑉𝐻𝑊))
4		simpr 490	. . . . . . 7 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → ¬ 𝑉 = 𝑊)
5	1, 3, 4	parhomne 50960	. . . . . 6 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → (𝑉 = 𝑋 ∧ 𝑊 = 𝑌))
6	5	simpld 500	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → 𝑉 = 𝑋)
7		parhomeqor.f	. . . . . . . . 9 ⊢ (𝜑 → 𝐹 ∈ (𝑈𝐻𝑉))
8	7	adantr 486	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝐹 ∈ (𝑈𝐻𝑉))
9		simpr 490	. . . . . . . 8 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → ¬ 𝑈 = 𝑉)
10	1, 8, 9	parhomne 50960	. . . . . . 7 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → (𝑈 = 𝑋 ∧ 𝑉 = 𝑌))
11	10	simprd 501	. . . . . 6 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝑉 = 𝑌)
12	11	adantr 486	. . . . 5 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → 𝑉 = 𝑌)
13	6, 12	eqtr3d 2800	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → 𝑋 = 𝑌)
14		parhomeqor2.n	. . . . . 6 ⊢ 𝑋 ≠ 𝑌
15	14	neii 2961	. . . . 5 ⊢ ¬ 𝑋 = 𝑌
16	15	a1i 11	. . . 4 ⊢ (((𝜑 ∧ ¬ 𝑈 = 𝑉) ∧ ¬ 𝑉 = 𝑊) → ¬ 𝑋 = 𝑌)
17	13, 16	condan 830	. . 3 ⊢ ((𝜑 ∧ ¬ 𝑈 = 𝑉) → 𝑉 = 𝑊)
18	17	ex 418	. 2 ⊢ (𝜑 → (¬ 𝑈 = 𝑉 → 𝑉 = 𝑊))
19	18	orrd 877	1 ⊢ (𝜑 → (𝑈 = 𝑉 ∨ 𝑉 = 𝑊))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ≠ wne 2959  Vcvv 3454  ∅c0 4281  ifcif 4485  {csn 4588  (class class class)co 7443   ∈ cmpo 7445

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5252  ax-nul 5263  ax-pr 5395

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-dm 5665  df-iota 6512  df-fv 6566  df-ov 7446  df-oprab 7447  df-mpo 7448

This theorem is used by: (None)