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Theorem elpr2elpr 4798
 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 ((𝑋𝑉𝑌𝑉𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
Distinct variable groups:   𝐴,𝑏   𝑉,𝑏   𝑋,𝑏   𝑌,𝑏

Proof of Theorem elpr2elpr
StepHypRef Expression
1 simprr 769 . . . . . 6 ((𝐴 = 𝑋 ∧ (𝑋𝑉𝑌𝑉)) → 𝑌𝑉)
2 preq2 4669 . . . . . . . 8 (𝑏 = 𝑌 → {𝐴, 𝑏} = {𝐴, 𝑌})
32eqeq2d 2837 . . . . . . 7 (𝑏 = 𝑌 → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑌}))
43adantl 482 . . . . . 6 (((𝐴 = 𝑋 ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏 = 𝑌) → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑌}))
5 preq1 4668 . . . . . . . 8 (𝑋 = 𝐴 → {𝑋, 𝑌} = {𝐴, 𝑌})
65eqcoms 2834 . . . . . . 7 (𝐴 = 𝑋 → {𝑋, 𝑌} = {𝐴, 𝑌})
76adantr 481 . . . . . 6 ((𝐴 = 𝑋 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} = {𝐴, 𝑌})
81, 4, 7rspcedvd 3630 . . . . 5 ((𝐴 = 𝑋 ∧ (𝑋𝑉𝑌𝑉)) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
98ex 413 . . . 4 (𝐴 = 𝑋 → ((𝑋𝑉𝑌𝑉) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
10 simprl 767 . . . . . 6 ((𝐴 = 𝑌 ∧ (𝑋𝑉𝑌𝑉)) → 𝑋𝑉)
11 preq2 4669 . . . . . . . 8 (𝑏 = 𝑋 → {𝐴, 𝑏} = {𝐴, 𝑋})
1211eqeq2d 2837 . . . . . . 7 (𝑏 = 𝑋 → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑋}))
1312adantl 482 . . . . . 6 (((𝐴 = 𝑌 ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏 = 𝑋) → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑋}))
14 preq2 4669 . . . . . . . . 9 (𝑌 = 𝐴 → {𝑋, 𝑌} = {𝑋, 𝐴})
1514eqcoms 2834 . . . . . . . 8 (𝐴 = 𝑌 → {𝑋, 𝑌} = {𝑋, 𝐴})
16 prcom 4667 . . . . . . . 8 {𝑋, 𝐴} = {𝐴, 𝑋}
1715, 16syl6eq 2877 . . . . . . 7 (𝐴 = 𝑌 → {𝑋, 𝑌} = {𝐴, 𝑋})
1817adantr 481 . . . . . 6 ((𝐴 = 𝑌 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} = {𝐴, 𝑋})
1910, 13, 18rspcedvd 3630 . . . . 5 ((𝐴 = 𝑌 ∧ (𝑋𝑉𝑌𝑉)) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
2019ex 413 . . . 4 (𝐴 = 𝑌 → ((𝑋𝑉𝑌𝑉) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
219, 20jaoi 853 . . 3 ((𝐴 = 𝑋𝐴 = 𝑌) → ((𝑋𝑉𝑌𝑉) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
22 elpri 4586 . . 3 (𝐴 ∈ {𝑋, 𝑌} → (𝐴 = 𝑋𝐴 = 𝑌))
2321, 22syl11 33 . 2 ((𝑋𝑉𝑌𝑉) → (𝐴 ∈ {𝑋, 𝑌} → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
24233impia 1111 1 ((𝑋𝑉𝑌𝑉𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 843   ∧ w3a 1081   = wceq 1530   ∈ wcel 2107  ∃wrex 3144  {cpr 4566 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798 This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-v 3502  df-un 3945  df-sn 4565  df-pr 4567 This theorem is referenced by:  upgredg2vtx  26859
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