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Theorem djuen 7239
Description: Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)
Assertion
Ref Expression
djuen ((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ≈ (𝐵𝐷))

Proof of Theorem djuen
StepHypRef Expression
1 encv 6771 . . . . . . . 8 (𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21adantr 276 . . . . . . 7 ((𝐴𝐵𝐶𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32simpld 112 . . . . . 6 ((𝐴𝐵𝐶𝐷) → 𝐴 ∈ V)
4 eninl 7125 . . . . . 6 (𝐴 ∈ V → (inl “ 𝐴) ≈ 𝐴)
53, 4syl 14 . . . . 5 ((𝐴𝐵𝐶𝐷) → (inl “ 𝐴) ≈ 𝐴)
6 simpl 109 . . . . 5 ((𝐴𝐵𝐶𝐷) → 𝐴𝐵)
7 entr 6809 . . . . 5 (((inl “ 𝐴) ≈ 𝐴𝐴𝐵) → (inl “ 𝐴) ≈ 𝐵)
85, 6, 7syl2anc 411 . . . 4 ((𝐴𝐵𝐶𝐷) → (inl “ 𝐴) ≈ 𝐵)
9 eninl 7125 . . . . . 6 (𝐵 ∈ V → (inl “ 𝐵) ≈ 𝐵)
102, 9simpl2im 386 . . . . 5 ((𝐴𝐵𝐶𝐷) → (inl “ 𝐵) ≈ 𝐵)
1110ensymd 6808 . . . 4 ((𝐴𝐵𝐶𝐷) → 𝐵 ≈ (inl “ 𝐵))
12 entr 6809 . . . 4 (((inl “ 𝐴) ≈ 𝐵𝐵 ≈ (inl “ 𝐵)) → (inl “ 𝐴) ≈ (inl “ 𝐵))
138, 11, 12syl2anc 411 . . 3 ((𝐴𝐵𝐶𝐷) → (inl “ 𝐴) ≈ (inl “ 𝐵))
14 encv 6771 . . . . . . . 8 (𝐶𝐷 → (𝐶 ∈ V ∧ 𝐷 ∈ V))
1514adantl 277 . . . . . . 7 ((𝐴𝐵𝐶𝐷) → (𝐶 ∈ V ∧ 𝐷 ∈ V))
1615simpld 112 . . . . . 6 ((𝐴𝐵𝐶𝐷) → 𝐶 ∈ V)
17 eninr 7126 . . . . . 6 (𝐶 ∈ V → (inr “ 𝐶) ≈ 𝐶)
1816, 17syl 14 . . . . 5 ((𝐴𝐵𝐶𝐷) → (inr “ 𝐶) ≈ 𝐶)
19 entr 6809 . . . . 5 (((inr “ 𝐶) ≈ 𝐶𝐶𝐷) → (inr “ 𝐶) ≈ 𝐷)
2018, 19sylancom 420 . . . 4 ((𝐴𝐵𝐶𝐷) → (inr “ 𝐶) ≈ 𝐷)
21 eninr 7126 . . . . . 6 (𝐷 ∈ V → (inr “ 𝐷) ≈ 𝐷)
2215, 21simpl2im 386 . . . . 5 ((𝐴𝐵𝐶𝐷) → (inr “ 𝐷) ≈ 𝐷)
2322ensymd 6808 . . . 4 ((𝐴𝐵𝐶𝐷) → 𝐷 ≈ (inr “ 𝐷))
24 entr 6809 . . . 4 (((inr “ 𝐶) ≈ 𝐷𝐷 ≈ (inr “ 𝐷)) → (inr “ 𝐶) ≈ (inr “ 𝐷))
2520, 23, 24syl2anc 411 . . 3 ((𝐴𝐵𝐶𝐷) → (inr “ 𝐶) ≈ (inr “ 𝐷))
26 djuin 7092 . . . 4 ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅
2726a1i 9 . . 3 ((𝐴𝐵𝐶𝐷) → ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅)
28 djuin 7092 . . . 4 ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅
2928a1i 9 . . 3 ((𝐴𝐵𝐶𝐷) → ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅)
30 unen 6841 . . 3 ((((inl “ 𝐴) ≈ (inl “ 𝐵) ∧ (inr “ 𝐶) ≈ (inr “ 𝐷)) ∧ (((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅ ∧ ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅)) → ((inl “ 𝐴) ∪ (inr “ 𝐶)) ≈ ((inl “ 𝐵) ∪ (inr “ 𝐷)))
3113, 25, 27, 29, 30syl22anc 1250 . 2 ((𝐴𝐵𝐶𝐷) → ((inl “ 𝐴) ∪ (inr “ 𝐶)) ≈ ((inl “ 𝐵) ∪ (inr “ 𝐷)))
32 djuun 7095 . 2 ((inl “ 𝐴) ∪ (inr “ 𝐶)) = (𝐴𝐶)
33 djuun 7095 . 2 ((inl “ 𝐵) ∪ (inr “ 𝐷)) = (𝐵𝐷)
3431, 32, 333brtr3g 4051 1 ((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ≈ (𝐵𝐷))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1364  wcel 2160  Vcvv 2752  cun 3142  cin 3143  c0 3437   class class class wbr 4018  cima 4647  cen 6763  cdju 7065  inlcinl 7073  inrcinr 7074
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-coll 4133  ax-sep 4136  ax-nul 4144  ax-pow 4192  ax-pr 4227  ax-un 4451
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-csb 3073  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-iun 3903  df-br 4019  df-opab 4080  df-mpt 4081  df-tr 4117  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1st 6164  df-2nd 6165  df-1o 6440  df-er 6558  df-en 6766  df-dju 7066  df-inl 7075  df-inr 7076
This theorem is referenced by:  djuenun  7240  exmidunben  12476  enctlem  12482
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