Theorem elpr2elpr 4759

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 772	. . . . . 6 ⊢ ((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → 𝑌 ∈ 𝑉)
2		preq2 4630	. . . . . . . 8 ⊢ (𝑏 = 𝑌 → {𝐴, 𝑏} = {𝐴, 𝑌})
3	2	eqeq2d 2809	. . . . . . 7 ⊢ (𝑏 = 𝑌 → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑌}))
4	3	adantl 485	. . . . . 6 ⊢ (((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑏 = 𝑌) → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑌}))
5		preq1 4629	. . . . . . . 8 ⊢ (𝑋 = 𝐴 → {𝑋, 𝑌} = {𝐴, 𝑌})
6	5	eqcoms 2806	. . . . . . 7 ⊢ (𝐴 = 𝑋 → {𝑋, 𝑌} = {𝐴, 𝑌})
7	6	adantr 484	. . . . . 6 ⊢ ((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → {𝑋, 𝑌} = {𝐴, 𝑌})
8	1, 4, 7	rspcedvd 3574	. . . . 5 ⊢ ((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
9	8	ex 416	. . . 4 ⊢ (𝐴 = 𝑋 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
10		simprl 770	. . . . . 6 ⊢ ((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → 𝑋 ∈ 𝑉)
11		preq2 4630	. . . . . . . 8 ⊢ (𝑏 = 𝑋 → {𝐴, 𝑏} = {𝐴, 𝑋})
12	11	eqeq2d 2809	. . . . . . 7 ⊢ (𝑏 = 𝑋 → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑋}))
13	12	adantl 485	. . . . . 6 ⊢ (((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑏 = 𝑋) → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑋}))
14		preq2 4630	. . . . . . . . 9 ⊢ (𝑌 = 𝐴 → {𝑋, 𝑌} = {𝑋, 𝐴})
15	14	eqcoms 2806	. . . . . . . 8 ⊢ (𝐴 = 𝑌 → {𝑋, 𝑌} = {𝑋, 𝐴})
16		prcom 4628	. . . . . . . 8 ⊢ {𝑋, 𝐴} = {𝐴, 𝑋}
17	15, 16	eqtrdi 2849	. . . . . . 7 ⊢ (𝐴 = 𝑌 → {𝑋, 𝑌} = {𝐴, 𝑋})
18	17	adantr 484	. . . . . 6 ⊢ ((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → {𝑋, 𝑌} = {𝐴, 𝑋})
19	10, 13, 18	rspcedvd 3574	. . . . 5 ⊢ ((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
20	19	ex 416	. . . 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 4547	. . 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 1114	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111  ∃wrex 3107  {cpr 4527

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-ral 3111  df-rex 3112  df-v 3443  df-un 3886  df-sn 4526  df-pr 4528

This theorem is referenced by:  upgredg2vtx  26934