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Theorem sprssspr 47406
Description: The set of all unordered pairs over a given set 𝑉 is a subset of the set of all unordered pairs. (Contributed by AV, 21-Nov-2021.)
Assertion
Ref Expression
sprssspr (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
Distinct variable group:   𝑉,𝑎,𝑏,𝑝

Proof of Theorem sprssspr
StepHypRef Expression
1 sprval 47404 . . 3 (𝑉 ∈ V → (Pairs‘𝑉) = {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}})
2 r2ex 3194 . . . . . . 7 (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} ↔ ∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝑝 = {𝑎, 𝑏}))
3 simpr 484 . . . . . . . 8 (((𝑎𝑉𝑏𝑉) ∧ 𝑝 = {𝑎, 𝑏}) → 𝑝 = {𝑎, 𝑏})
432eximi 1833 . . . . . . 7 (∃𝑎𝑏((𝑎𝑉𝑏𝑉) ∧ 𝑝 = {𝑎, 𝑏}) → ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
52, 4sylbi 217 . . . . . 6 (∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} → ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
65ax-gen 1792 . . . . 5 𝑝(∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} → ∃𝑎𝑏 𝑝 = {𝑎, 𝑏})
76a1i 11 . . . 4 (𝑉 ∈ V → ∀𝑝(∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} → ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
8 ss2ab 4072 . . . 4 ({𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}} ↔ ∀𝑝(∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏} → ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}))
97, 8sylibr 234 . . 3 (𝑉 ∈ V → {𝑝 ∣ ∃𝑎𝑉𝑏𝑉 𝑝 = {𝑎, 𝑏}} ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
101, 9eqsstrd 4034 . 2 (𝑉 ∈ V → (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
11 fvprc 6899 . . 3 𝑉 ∈ V → (Pairs‘𝑉) = ∅)
12 0ss 4406 . . . 4 ∅ ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
1312a1i 11 . . 3 𝑉 ∈ V → ∅ ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
1411, 13eqsstrd 4034 . 2 𝑉 ∈ V → (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}})
1510, 14pm2.61i 182 1 (Pairs‘𝑉) ⊆ {𝑝 ∣ ∃𝑎𝑏 𝑝 = {𝑎, 𝑏}}
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 395  wal 1535   = wceq 1537  wex 1776  wcel 2106  {cab 2712  wrex 3068  Vcvv 3478  wss 3963  c0 4339  {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-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:  spr0el  47407
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