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Theorem elpr2elpr 4429
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 811 . . . . . 6 ((𝐴 = 𝑋 ∧ (𝑋𝑉𝑌𝑉)) → 𝑌𝑉)
2 preq2 4301 . . . . . . . 8 (𝑏 = 𝑌 → {𝐴, 𝑏} = {𝐴, 𝑌})
32eqeq2d 2661 . . . . . . 7 (𝑏 = 𝑌 → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑌}))
43adantl 481 . . . . . 6 (((𝐴 = 𝑋 ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏 = 𝑌) → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑌}))
5 preq1 4300 . . . . . . . 8 (𝑋 = 𝐴 → {𝑋, 𝑌} = {𝐴, 𝑌})
65eqcoms 2659 . . . . . . 7 (𝐴 = 𝑋 → {𝑋, 𝑌} = {𝐴, 𝑌})
76adantr 480 . . . . . 6 ((𝐴 = 𝑋 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} = {𝐴, 𝑌})
81, 4, 7rspcedvd 3348 . . . . 5 ((𝐴 = 𝑋 ∧ (𝑋𝑉𝑌𝑉)) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
98ex 449 . . . 4 (𝐴 = 𝑋 → ((𝑋𝑉𝑌𝑉) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
10 simprl 809 . . . . . 6 ((𝐴 = 𝑌 ∧ (𝑋𝑉𝑌𝑉)) → 𝑋𝑉)
11 preq2 4301 . . . . . . . 8 (𝑏 = 𝑋 → {𝐴, 𝑏} = {𝐴, 𝑋})
1211eqeq2d 2661 . . . . . . 7 (𝑏 = 𝑋 → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑋}))
1312adantl 481 . . . . . 6 (((𝐴 = 𝑌 ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏 = 𝑋) → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑋}))
14 preq2 4301 . . . . . . . . 9 (𝑌 = 𝐴 → {𝑋, 𝑌} = {𝑋, 𝐴})
1514eqcoms 2659 . . . . . . . 8 (𝐴 = 𝑌 → {𝑋, 𝑌} = {𝑋, 𝐴})
16 prcom 4299 . . . . . . . 8 {𝑋, 𝐴} = {𝐴, 𝑋}
1715, 16syl6eq 2701 . . . . . . 7 (𝐴 = 𝑌 → {𝑋, 𝑌} = {𝐴, 𝑋})
1817adantr 480 . . . . . 6 ((𝐴 = 𝑌 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} = {𝐴, 𝑋})
1910, 13, 18rspcedvd 3348 . . . . 5 ((𝐴 = 𝑌 ∧ (𝑋𝑉𝑌𝑉)) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
2019ex 449 . . . 4 (𝐴 = 𝑌 → ((𝑋𝑉𝑌𝑉) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
219, 20jaoi 393 . . 3 ((𝐴 = 𝑋𝐴 = 𝑌) → ((𝑋𝑉𝑌𝑉) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
22 elpri 4230 . . 3 (𝐴 ∈ {𝑋, 𝑌} → (𝐴 = 𝑋𝐴 = 𝑌))
2321, 22syl11 33 . 2 ((𝑋𝑉𝑌𝑉) → (𝐴 ∈ {𝑋, 𝑌} → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
24233impia 1280 1 ((𝑋𝑉𝑌𝑉𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 382  wa 383  w3a 1054   = wceq 1523  wcel 2030  wrex 2942  {cpr 4212
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-un 3612  df-sn 4211  df-pr 4213
This theorem is referenced by:  upgredg2vtx  26081
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