Theorem elpr2elpr 4826

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 773	. . . . . 6 ⊢ ((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → 𝑌 ∈ 𝑉)
2		preq12 4693	. . . . . . . 8 ⊢ ((𝐴 = 𝑋 ∧ 𝑏 = 𝑌) → {𝐴, 𝑏} = {𝑋, 𝑌})
3	2	eqcomd 2743	. . . . . . 7 ⊢ ((𝐴 = 𝑋 ∧ 𝑏 = 𝑌) → {𝑋, 𝑌} = {𝐴, 𝑏})
4	3	adantlr 716	. . . . . 6 ⊢ (((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑏 = 𝑌) → {𝑋, 𝑌} = {𝐴, 𝑏})
5	1, 4	rspcedeq2vd 3585	. . . . 5 ⊢ ((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
6	5	ex 412	. . . 4 ⊢ (𝐴 = 𝑋 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
7		simprl 771	. . . . . 6 ⊢ ((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → 𝑋 ∈ 𝑉)
8		preq12 4693	. . . . . . . 8 ⊢ ((𝐴 = 𝑌 ∧ 𝑏 = 𝑋) → {𝐴, 𝑏} = {𝑌, 𝑋})
9		prcom 4690	. . . . . . . 8 ⊢ {𝑌, 𝑋} = {𝑋, 𝑌}
10	8, 9	eqtr2di 2789	. . . . . . 7 ⊢ ((𝐴 = 𝑌 ∧ 𝑏 = 𝑋) → {𝑋, 𝑌} = {𝐴, 𝑏})
11	10	adantlr 716	. . . . . 6 ⊢ (((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑏 = 𝑋) → {𝑋, 𝑌} = {𝐴, 𝑏})
12	7, 11	rspcedeq2vd 3585	. . . . 5 ⊢ ((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
13	12	ex 412	. . . 4 ⊢ (𝐴 = 𝑌 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
14	6, 13	jaoi 858	. . 3 ⊢ ((𝐴 = 𝑋 ∨ 𝐴 = 𝑌) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
15		elpri 4605	. . 3 ⊢ (𝐴 ∈ {𝑋, 𝑌} → (𝐴 = 𝑋 ∨ 𝐴 = 𝑌))
16	14, 15	syl11 33	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐴 ∈ {𝑋, 𝑌} → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
17	16	3impia 1118	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 848   ∧ w3a 1087   = wceq 1542   ∈ wcel 2114  ∃wrex 3061  {cpr 4583

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-v 3443  df-un 3907  df-sn 4582  df-pr 4584

This theorem is referenced by:  upgredg2vtx  29197