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Theorem qdass 4442
Description: Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.)
Assertion
Ref Expression
qdass ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴, 𝐵, 𝐶} ∪ {𝐷})

Proof of Theorem qdass
StepHypRef Expression
1 unass 3931 . 2 (({𝐴, 𝐵} ∪ {𝐶}) ∪ {𝐷}) = ({𝐴, 𝐵} ∪ ({𝐶} ∪ {𝐷}))
2 df-tp 4338 . . 3 {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶})
32uneq1i 3924 . 2 ({𝐴, 𝐵, 𝐶} ∪ {𝐷}) = (({𝐴, 𝐵} ∪ {𝐶}) ∪ {𝐷})
4 df-pr 4336 . . 3 {𝐶, 𝐷} = ({𝐶} ∪ {𝐷})
54uneq2i 3925 . 2 ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴, 𝐵} ∪ ({𝐶} ∪ {𝐷}))
61, 3, 53eqtr4ri 2797 1 ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴, 𝐵, 𝐶} ∪ {𝐷})
Colors of variables: wff setvar class
Syntax hints:   = wceq 1652  cun 3729  {csn 4333  {cpr 4335  {ctp 4337
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2069  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2742
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2062  df-clab 2751  df-cleq 2757  df-clel 2760  df-nfc 2895  df-v 3351  df-un 3736  df-pr 4336  df-tp 4338
This theorem is referenced by:  cnlmodlem1  23213  cnlmodlem2  23214  cnlmodlem3  23215  cnlmod4  23216  cnstrcvs  23218  ex-pw  27679
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