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Theorem en2prd 9031
Description: Two unordered pairs are equinumerous. (Contributed by BTernaryTau, 23-Dec-2024.)
Hypotheses
Ref Expression
en2prd.1 (𝜑𝐴𝑉)
en2prd.2 (𝜑𝐵𝑊)
en2prd.3 (𝜑𝐶𝑋)
en2prd.4 (𝜑𝐷𝑌)
en2prd.5 (𝜑𝐴𝐵)
en2prd.6 (𝜑𝐶𝐷)
Assertion
Ref Expression
en2prd (𝜑 → {𝐴, 𝐵} ≈ {𝐶, 𝐷})

Proof of Theorem en2prd
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 prex 5425 . . 3 {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ∈ V
2 en2prd.5 . . . 4 (𝜑𝐴𝐵)
3 en2prd.6 . . . 4 (𝜑𝐶𝐷)
4 en2prd.1 . . . . 5 (𝜑𝐴𝑉)
5 en2prd.3 . . . . 5 (𝜑𝐶𝑋)
6 en2prd.2 . . . . 5 (𝜑𝐵𝑊)
7 en2prd.4 . . . . 5 (𝜑𝐷𝑌)
8 f1oprg 6865 . . . . 5 (((𝐴𝑉𝐶𝑋) ∧ (𝐵𝑊𝐷𝑌)) → ((𝐴𝐵𝐶𝐷) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
94, 5, 6, 7, 8syl22anc 837 . . . 4 (𝜑 → ((𝐴𝐵𝐶𝐷) → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
102, 3, 9mp2and 697 . . 3 (𝜑 → {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
11 f1oeq1 6808 . . . 4 (𝑓 = {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} → (𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷} ↔ {⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
1211spcegv 3584 . . 3 ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩} ∈ V → ({⟨𝐴, 𝐶⟩, ⟨𝐵, 𝐷⟩}:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷} → ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
131, 10, 12mpsyl 68 . 2 (𝜑 → ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
14 prex 5425 . . 3 {𝐴, 𝐵} ∈ V
15 prex 5425 . . 3 {𝐶, 𝐷} ∈ V
16 breng 8931 . . 3 (({𝐴, 𝐵} ∈ V ∧ {𝐶, 𝐷} ∈ V) → ({𝐴, 𝐵} ≈ {𝐶, 𝐷} ↔ ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷}))
1714, 15, 16mp2an 690 . 2 ({𝐴, 𝐵} ≈ {𝐶, 𝐷} ↔ ∃𝑓 𝑓:{𝐴, 𝐵}–1-1-onto→{𝐶, 𝐷})
1813, 17sylibr 233 1 (𝜑 → {𝐴, 𝐵} ≈ {𝐶, 𝐷})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wex 1781  wcel 2106  wne 2939  Vcvv 3473  {cpr 4624  cop 4628   class class class wbr 5141  1-1-ontowf1o 6531  cen 8919
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2702  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2533  df-clab 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4523  df-sn 4623  df-pr 4625  df-op 4629  df-br 5142  df-opab 5204  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-fun 6534  df-fn 6535  df-f 6536  df-f1 6537  df-fo 6538  df-f1o 6539  df-en 8923
This theorem is referenced by:  enpr2d  9032  rex2dom  9229
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