Theorem elpr2elpr 4810

Description: For an element 𝐴 of an unordered pair which is a subset of a given set 𝑉, there is another (maybe the same) element 𝑏 of the given set 𝑉 being an element of the unordered pair. (Contributed by AV, 5-Dec-2020.)

Assertion

Ref	Expression
elpr2elpr	⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})

Distinct variable groups:   𝐴,𝑏   𝑉,𝑏   𝑋,𝑏   𝑌,𝑏

Proof of Theorem elpr2elpr

Step	Hyp	Ref	Expression
1		simprr 770	. . . . . 6 ⊢ ((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → 𝑌 ∈ 𝑉)
2		preq2 4679	. . . . . . . 8 ⊢ (𝑏 = 𝑌 → {𝐴, 𝑏} = {𝐴, 𝑌})
3	2	eqeq2d 2747	. . . . . . 7 ⊢ (𝑏 = 𝑌 → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑌}))
4	3	adantl 482	. . . . . 6 ⊢ (((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑏 = 𝑌) → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑌}))
5		preq1 4678	. . . . . . . 8 ⊢ (𝑋 = 𝐴 → {𝑋, 𝑌} = {𝐴, 𝑌})
6	5	eqcoms 2744	. . . . . . 7 ⊢ (𝐴 = 𝑋 → {𝑋, 𝑌} = {𝐴, 𝑌})
7	6	adantr 481	. . . . . 6 ⊢ ((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → {𝑋, 𝑌} = {𝐴, 𝑌})
8	1, 4, 7	rspcedvd 3571	. . . . 5 ⊢ ((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
9	8	ex 413	. . . 4 ⊢ (𝐴 = 𝑋 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
10		simprl 768	. . . . . 6 ⊢ ((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → 𝑋 ∈ 𝑉)
11		preq2 4679	. . . . . . . 8 ⊢ (𝑏 = 𝑋 → {𝐴, 𝑏} = {𝐴, 𝑋})
12	11	eqeq2d 2747	. . . . . . 7 ⊢ (𝑏 = 𝑋 → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑋}))
13	12	adantl 482	. . . . . 6 ⊢ (((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑏 = 𝑋) → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑋}))
14		preq2 4679	. . . . . . . . 9 ⊢ (𝑌 = 𝐴 → {𝑋, 𝑌} = {𝑋, 𝐴})
15	14	eqcoms 2744	. . . . . . . 8 ⊢ (𝐴 = 𝑌 → {𝑋, 𝑌} = {𝑋, 𝐴})
16		prcom 4677	. . . . . . . 8 ⊢ {𝑋, 𝐴} = {𝐴, 𝑋}
17	15, 16	eqtrdi 2792	. . . . . . 7 ⊢ (𝐴 = 𝑌 → {𝑋, 𝑌} = {𝐴, 𝑋})
18	17	adantr 481	. . . . . 6 ⊢ ((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → {𝑋, 𝑌} = {𝐴, 𝑋})
19	10, 13, 18	rspcedvd 3571	. . . . 5 ⊢ ((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
20	19	ex 413	. . . 4 ⊢ (𝐴 = 𝑌 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
21	9, 20	jaoi 854	. . 3 ⊢ ((𝐴 = 𝑋 ∨ 𝐴 = 𝑌) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
22		elpri 4592	. . 3 ⊢ (𝐴 ∈ {𝑋, 𝑌} → (𝐴 = 𝑋 ∨ 𝐴 = 𝑌))
23	21, 22	syl11 33	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐴 ∈ {𝑋, 𝑌} → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
24	23	3impia 1116	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1540   ∈ wcel 2105  ∃wrex 3070  {cpr 4572

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-ex 1781  df-sb 2067  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3062  df-rex 3071  df-v 3442  df-un 3901  df-sn 4571  df-pr 4573

This theorem is referenced by:  upgredg2vtx  27644