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Theorem elpr2elpr 4336
 Description: For an element of an unordered pair which is a subset of a given set, there is another (maybe the same) element of the given set 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 elpri 4145 . . . 4 (𝐴 ∈ {𝑋, 𝑌} → (𝐴 = 𝑋𝐴 = 𝑌))
2 simprr 792 . . . . . . 7 ((𝐴 = 𝑋 ∧ (𝑋𝑉𝑌𝑉)) → 𝑌𝑉)
3 preq2 4213 . . . . . . . . 9 (𝑏 = 𝑌 → {𝐴, 𝑏} = {𝐴, 𝑌})
43eqeq2d 2620 . . . . . . . 8 (𝑏 = 𝑌 → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑌}))
54adantl 481 . . . . . . 7 (((𝐴 = 𝑋 ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏 = 𝑌) → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑌}))
6 preq1 4212 . . . . . . . . 9 (𝑋 = 𝐴 → {𝑋, 𝑌} = {𝐴, 𝑌})
76eqcoms 2618 . . . . . . . 8 (𝐴 = 𝑋 → {𝑋, 𝑌} = {𝐴, 𝑌})
87adantr 480 . . . . . . 7 ((𝐴 = 𝑋 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} = {𝐴, 𝑌})
92, 5, 8rspcedvd 3289 . . . . . 6 ((𝐴 = 𝑋 ∧ (𝑋𝑉𝑌𝑉)) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
109ex 449 . . . . 5 (𝐴 = 𝑋 → ((𝑋𝑉𝑌𝑉) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
11 simprl 790 . . . . . . 7 ((𝐴 = 𝑌 ∧ (𝑋𝑉𝑌𝑉)) → 𝑋𝑉)
12 preq2 4213 . . . . . . . . 9 (𝑏 = 𝑋 → {𝐴, 𝑏} = {𝐴, 𝑋})
1312eqeq2d 2620 . . . . . . . 8 (𝑏 = 𝑋 → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑋}))
1413adantl 481 . . . . . . 7 (((𝐴 = 𝑌 ∧ (𝑋𝑉𝑌𝑉)) ∧ 𝑏 = 𝑋) → ({𝑋, 𝑌} = {𝐴, 𝑏} ↔ {𝑋, 𝑌} = {𝐴, 𝑋}))
15 preq2 4213 . . . . . . . . . 10 (𝑌 = 𝐴 → {𝑋, 𝑌} = {𝑋, 𝐴})
1615eqcoms 2618 . . . . . . . . 9 (𝐴 = 𝑌 → {𝑋, 𝑌} = {𝑋, 𝐴})
17 prcom 4211 . . . . . . . . 9 {𝑋, 𝐴} = {𝐴, 𝑋}
1816, 17syl6eq 2660 . . . . . . . 8 (𝐴 = 𝑌 → {𝑋, 𝑌} = {𝐴, 𝑋})
1918adantr 480 . . . . . . 7 ((𝐴 = 𝑌 ∧ (𝑋𝑉𝑌𝑉)) → {𝑋, 𝑌} = {𝐴, 𝑋})
2011, 14, 19rspcedvd 3289 . . . . . 6 ((𝐴 = 𝑌 ∧ (𝑋𝑉𝑌𝑉)) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
2120ex 449 . . . . 5 (𝐴 = 𝑌 → ((𝑋𝑉𝑌𝑉) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
2210, 21jaoi 393 . . . 4 ((𝐴 = 𝑋𝐴 = 𝑌) → ((𝑋𝑉𝑌𝑉) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
231, 22syl 17 . . 3 (𝐴 ∈ {𝑋, 𝑌} → ((𝑋𝑉𝑌𝑉) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
2423com12 32 . 2 ((𝑋𝑉𝑌𝑉) → (𝐴 ∈ {𝑋, 𝑌} → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏}))
25243impia 1253 1 ((𝑋𝑉𝑌𝑉𝐴 ∈ {𝑋, 𝑌}) → ∃𝑏𝑉 {𝑋, 𝑌} = {𝐴, 𝑏})
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   = wceq 1475   ∈ wcel 1977  ∃wrex 2897  {cpr 4127 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-v 3175  df-un 3545  df-sn 4126  df-pr 4128 This theorem is referenced by:  upgredg2vtx  25814
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