Theorem elpr2elpr 3825

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 531	. . . . . 6 ⊢ ((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → 𝑌 ∈ 𝑉)
2		preq12 3717	. . . . . . . 8 ⊢ ((𝐴 = 𝑋 ∧ 𝑏 = 𝑌) → {𝐴, 𝑏} = {𝑋, 𝑌})
3	2	eqcomd 2212	. . . . . . 7 ⊢ ((𝐴 = 𝑋 ∧ 𝑏 = 𝑌) → {𝑋, 𝑌} = {𝐴, 𝑏})
4	3	adantlr 477	. . . . . 6 ⊢ (((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑏 = 𝑌) → {𝑋, 𝑌} = {𝐴, 𝑏})
5	1, 4	rspcedeq2vd 2891	. . . . 5 ⊢ ((𝐴 = 𝑋 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
6	5	ex 115	. . . 4 ⊢ (𝐴 = 𝑋 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
7		simprl 529	. . . . . 6 ⊢ ((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → 𝑋 ∈ 𝑉)
8		preq12 3717	. . . . . . . 8 ⊢ ((𝐴 = 𝑌 ∧ 𝑏 = 𝑋) → {𝐴, 𝑏} = {𝑌, 𝑋})
9		prcom 3714	. . . . . . . 8 ⊢ {𝑌, 𝑋} = {𝑋, 𝑌}
10	8, 9	eqtr2di 2256	. . . . . . 7 ⊢ ((𝐴 = 𝑌 ∧ 𝑏 = 𝑋) → {𝑋, 𝑌} = {𝐴, 𝑏})
11	10	adantlr 477	. . . . . 6 ⊢ (((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) ∧ 𝑏 = 𝑋) → {𝑋, 𝑌} = {𝐴, 𝑏})
12	7, 11	rspcedeq2vd 2891	. . . . 5 ⊢ ((𝐴 = 𝑌 ∧ (𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉)) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
13	12	ex 115	. . . 4 ⊢ (𝐴 = 𝑌 → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
14	6, 13	jaoi 718	. . 3 ⊢ ((𝐴 = 𝑋 ∨ 𝐴 = 𝑌) → ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
15		elpri 3661	. . 3 ⊢ (𝐴 ∈ {𝑋, 𝑌} → (𝐴 = 𝑋 ∨ 𝐴 = 𝑌))
16	14, 15	syl11 31	. 2 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉) → (𝐴 ∈ {𝑋, 𝑌} → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
17	16	3impia 1203	1 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑉 ∧ 𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏 ∈ 𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 710   ∧ w3a 981   = wceq 1373   ∈ wcel 2177  ∃wrex 2486  {cpr 3639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-rex 2491  df-v 2775  df-un 3174  df-sn 3644  df-pr 3645

This theorem is referenced by:  upgredg2vtx  15822