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Theorem endisj 6234
Description: Any two sets are equinumerous to disjoint sets. Exercise 4.39 of [Mendelson] p. 255. (Contributed by NM, 16-Apr-2004.)
Hypotheses
Ref Expression
endisj.1 A V
endisj.2 B V
Assertion
Ref Expression
endisj xy((xA yB) (xy) = ∅)
Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem endisj
StepHypRef Expression
1 endisj.1 . . . 4 A V
2 0ex 3875 . . . 4 V
31, 2xpsnen 6231 . . 3 (A × {∅}) ≈ A
4 endisj.2 . . . 4 B V
5 1on 5947 . . . . 5 1𝑜 On
65elexi 2561 . . . 4 1𝑜 V
74, 6xpsnen 6231 . . 3 (B × {1𝑜}) ≈ B
83, 7pm3.2i 257 . 2 ((A × {∅}) ≈ A (B × {1𝑜}) ≈ B)
9 xp01disj 5956 . 2 ((A × {∅}) ∩ (B × {1𝑜})) = ∅
10 p0ex 3930 . . . 4 {∅} V
111, 10xpex 4396 . . 3 (A × {∅}) V
126snex 3928 . . . 4 {1𝑜} V
134, 12xpex 4396 . . 3 (B × {1𝑜}) V
14 breq1 3758 . . . . 5 (x = (A × {∅}) → (xA ↔ (A × {∅}) ≈ A))
15 breq1 3758 . . . . 5 (y = (B × {1𝑜}) → (yB ↔ (B × {1𝑜}) ≈ B))
1614, 15bi2anan9 538 . . . 4 ((x = (A × {∅}) y = (B × {1𝑜})) → ((xA yB) ↔ ((A × {∅}) ≈ A (B × {1𝑜}) ≈ B)))
17 ineq12 3127 . . . . 5 ((x = (A × {∅}) y = (B × {1𝑜})) → (xy) = ((A × {∅}) ∩ (B × {1𝑜})))
1817eqeq1d 2045 . . . 4 ((x = (A × {∅}) y = (B × {1𝑜})) → ((xy) = ∅ ↔ ((A × {∅}) ∩ (B × {1𝑜})) = ∅))
1916, 18anbi12d 442 . . 3 ((x = (A × {∅}) y = (B × {1𝑜})) → (((xA yB) (xy) = ∅) ↔ (((A × {∅}) ≈ A (B × {1𝑜}) ≈ B) ((A × {∅}) ∩ (B × {1𝑜})) = ∅)))
2011, 13, 19spc2ev 2642 . 2 ((((A × {∅}) ≈ A (B × {1𝑜}) ≈ B) ((A × {∅}) ∩ (B × {1𝑜})) = ∅) → xy((xA yB) (xy) = ∅))
218, 9, 20mp2an 402 1 xy((xA yB) (xy) = ∅)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  Vcvv 2551  cin 2910  c0 3218  {csn 3367   class class class wbr 3755  Oncon0 4066   × cxp 4286  1𝑜c1o 5933  cen 6155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-fal 1248  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ne 2203  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-int 3607  df-br 3756  df-opab 3810  df-mpt 3811  df-tr 3846  df-id 4021  df-iord 4069  df-on 4071  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-1o 5940  df-en 6158
This theorem is referenced by: (None)
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