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Theorem uniprg 4599
Description: The union of a pair is the union of its members. Proposition 5.7 of [TakeutiZaring] p. 16. (Contributed by NM, 25-Aug-2006.)
Assertion
Ref Expression
uniprg ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))

Proof of Theorem uniprg
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 preq1 4415 . . . 4 (𝑥 = 𝐴 → {𝑥, 𝑦} = {𝐴, 𝑦})
21unieqd 4595 . . 3 (𝑥 = 𝐴 {𝑥, 𝑦} = {𝐴, 𝑦})
3 uneq1 3918 . . 3 (𝑥 = 𝐴 → (𝑥𝑦) = (𝐴𝑦))
42, 3eqeq12d 2789 . 2 (𝑥 = 𝐴 → ( {𝑥, 𝑦} = (𝑥𝑦) ↔ {𝐴, 𝑦} = (𝐴𝑦)))
5 preq2 4416 . . . 4 (𝑦 = 𝐵 → {𝐴, 𝑦} = {𝐴, 𝐵})
65unieqd 4595 . . 3 (𝑦 = 𝐵 {𝐴, 𝑦} = {𝐴, 𝐵})
7 uneq2 3919 . . 3 (𝑦 = 𝐵 → (𝐴𝑦) = (𝐴𝐵))
86, 7eqeq12d 2789 . 2 (𝑦 = 𝐵 → ( {𝐴, 𝑦} = (𝐴𝑦) ↔ {𝐴, 𝐵} = (𝐴𝐵)))
9 vex 3358 . . 3 𝑥 ∈ V
10 vex 3358 . . 3 𝑦 ∈ V
119, 10unipr 4598 . 2 {𝑥, 𝑦} = (𝑥𝑦)
124, 8, 11vtocl2g 3426 1 ((𝐴𝑉𝐵𝑊) → {𝐴, 𝐵} = (𝐴𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1634  wcel 2148  cun 3727  {cpr 4328   cuni 4585
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1873  ax-4 1888  ax-5 1994  ax-6 2060  ax-7 2096  ax-9 2157  ax-10 2177  ax-11 2193  ax-12 2206  ax-13 2411  ax-ext 2754
This theorem depends on definitions:  df-bi 198  df-an 384  df-or 864  df-tru 1637  df-ex 1856  df-nf 1861  df-sb 2053  df-clab 2761  df-cleq 2767  df-clel 2770  df-nfc 2905  df-rex 3070  df-v 3357  df-un 3734  df-sn 4327  df-pr 4329  df-uni 4586
This theorem is referenced by:  wunun  9755  tskun  9831  gruun  9851  mrcun  16510  unopn  20948  indistopon  21046  unconn  21473  limcun  23900  sshjval3  28570  prsiga  30551  unelsiga  30554  unelldsys  30578  measxun2  30630  measssd  30635  carsgsigalem  30734  carsgclctun  30740  pmeasmono  30743  probun  30838  indispconn  31571  kelac2  38176  fourierdlem70  40916  fourierdlem71  40917  saluncl  41060  prsal  41061  meadjun  41202  omeunle  41256
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