Users' Mathboxes Mathbox for Alexander van der Vekens < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  prsprel Structured version   Visualization version   GIF version

Theorem prsprel 47412
Description: The elements of a pair from the set of all unordered pairs over a given set 𝑉 are elements of the set 𝑉. (Contributed by AV, 22-Nov-2021.)
Assertion
Ref Expression
prsprel (({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋𝑈𝑌𝑊)) → (𝑋𝑉𝑌𝑉))

Proof of Theorem prsprel
Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sprel 47409 . . 3 ({𝑋, 𝑌} ∈ (Pairs‘𝑉) → ∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏})
2 preq12bg 4858 . . . . . . 7 (((𝑋𝑈𝑌𝑊) ∧ (𝑎𝑉𝑏𝑉)) → ({𝑋, 𝑌} = {𝑎, 𝑏} ↔ ((𝑋 = 𝑎𝑌 = 𝑏) ∨ (𝑋 = 𝑏𝑌 = 𝑎))))
3 eleq1 2827 . . . . . . . . . . . . 13 (𝑎 = 𝑋 → (𝑎𝑉𝑋𝑉))
43eqcoms 2743 . . . . . . . . . . . 12 (𝑋 = 𝑎 → (𝑎𝑉𝑋𝑉))
5 eleq1 2827 . . . . . . . . . . . . 13 (𝑏 = 𝑌 → (𝑏𝑉𝑌𝑉))
65eqcoms 2743 . . . . . . . . . . . 12 (𝑌 = 𝑏 → (𝑏𝑉𝑌𝑉))
74, 6bi2anan9 638 . . . . . . . . . . 11 ((𝑋 = 𝑎𝑌 = 𝑏) → ((𝑎𝑉𝑏𝑉) ↔ (𝑋𝑉𝑌𝑉)))
87biimpd 229 . . . . . . . . . 10 ((𝑋 = 𝑎𝑌 = 𝑏) → ((𝑎𝑉𝑏𝑉) → (𝑋𝑉𝑌𝑉)))
9 eleq1 2827 . . . . . . . . . . . . . 14 (𝑏 = 𝑋 → (𝑏𝑉𝑋𝑉))
109eqcoms 2743 . . . . . . . . . . . . 13 (𝑋 = 𝑏 → (𝑏𝑉𝑋𝑉))
11 eleq1 2827 . . . . . . . . . . . . . 14 (𝑎 = 𝑌 → (𝑎𝑉𝑌𝑉))
1211eqcoms 2743 . . . . . . . . . . . . 13 (𝑌 = 𝑎 → (𝑎𝑉𝑌𝑉))
1310, 12bi2anan9 638 . . . . . . . . . . . 12 ((𝑋 = 𝑏𝑌 = 𝑎) → ((𝑏𝑉𝑎𝑉) ↔ (𝑋𝑉𝑌𝑉)))
1413biimpd 229 . . . . . . . . . . 11 ((𝑋 = 𝑏𝑌 = 𝑎) → ((𝑏𝑉𝑎𝑉) → (𝑋𝑉𝑌𝑉)))
1514ancomsd 465 . . . . . . . . . 10 ((𝑋 = 𝑏𝑌 = 𝑎) → ((𝑎𝑉𝑏𝑉) → (𝑋𝑉𝑌𝑉)))
168, 15jaoi 857 . . . . . . . . 9 (((𝑋 = 𝑎𝑌 = 𝑏) ∨ (𝑋 = 𝑏𝑌 = 𝑎)) → ((𝑎𝑉𝑏𝑉) → (𝑋𝑉𝑌𝑉)))
1716com12 32 . . . . . . . 8 ((𝑎𝑉𝑏𝑉) → (((𝑋 = 𝑎𝑌 = 𝑏) ∨ (𝑋 = 𝑏𝑌 = 𝑎)) → (𝑋𝑉𝑌𝑉)))
1817adantl 481 . . . . . . 7 (((𝑋𝑈𝑌𝑊) ∧ (𝑎𝑉𝑏𝑉)) → (((𝑋 = 𝑎𝑌 = 𝑏) ∨ (𝑋 = 𝑏𝑌 = 𝑎)) → (𝑋𝑉𝑌𝑉)))
192, 18sylbid 240 . . . . . 6 (((𝑋𝑈𝑌𝑊) ∧ (𝑎𝑉𝑏𝑉)) → ({𝑋, 𝑌} = {𝑎, 𝑏} → (𝑋𝑉𝑌𝑉)))
2019expcom 413 . . . . 5 ((𝑎𝑉𝑏𝑉) → ((𝑋𝑈𝑌𝑊) → ({𝑋, 𝑌} = {𝑎, 𝑏} → (𝑋𝑉𝑌𝑉))))
2120com23 86 . . . 4 ((𝑎𝑉𝑏𝑉) → ({𝑋, 𝑌} = {𝑎, 𝑏} → ((𝑋𝑈𝑌𝑊) → (𝑋𝑉𝑌𝑉))))
2221rexlimivv 3199 . . 3 (∃𝑎𝑉𝑏𝑉 {𝑋, 𝑌} = {𝑎, 𝑏} → ((𝑋𝑈𝑌𝑊) → (𝑋𝑉𝑌𝑉)))
231, 22syl 17 . 2 ({𝑋, 𝑌} ∈ (Pairs‘𝑉) → ((𝑋𝑈𝑌𝑊) → (𝑋𝑉𝑌𝑉)))
2423imp 406 1 (({𝑋, 𝑌} ∈ (Pairs‘𝑉) ∧ (𝑋𝑈𝑌𝑊)) → (𝑋𝑉𝑌𝑉))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wo 847   = wceq 1537  wcel 2106  wrex 3068  {cpr 4633  cfv 6563  Pairscspr 47402
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fv 6571  df-spr 47403
This theorem is referenced by:  prsssprel  47413  sprsymrelfolem2  47418
  Copyright terms: Public domain W3C validator