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Theorem djuen 7083
 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 6647 . . . . . . . 8 (𝐴𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
21adantr 274 . . . . . . 7 ((𝐴𝐵𝐶𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
32simpld 111 . . . . . 6 ((𝐴𝐵𝐶𝐷) → 𝐴 ∈ V)
4 eninl 6989 . . . . . 6 (𝐴 ∈ V → (inl “ 𝐴) ≈ 𝐴)
53, 4syl 14 . . . . 5 ((𝐴𝐵𝐶𝐷) → (inl “ 𝐴) ≈ 𝐴)
6 simpl 108 . . . . 5 ((𝐴𝐵𝐶𝐷) → 𝐴𝐵)
7 entr 6685 . . . . 5 (((inl “ 𝐴) ≈ 𝐴𝐴𝐵) → (inl “ 𝐴) ≈ 𝐵)
85, 6, 7syl2anc 409 . . . 4 ((𝐴𝐵𝐶𝐷) → (inl “ 𝐴) ≈ 𝐵)
9 eninl 6989 . . . . . 6 (𝐵 ∈ V → (inl “ 𝐵) ≈ 𝐵)
102, 9simpl2im 384 . . . . 5 ((𝐴𝐵𝐶𝐷) → (inl “ 𝐵) ≈ 𝐵)
1110ensymd 6684 . . . 4 ((𝐴𝐵𝐶𝐷) → 𝐵 ≈ (inl “ 𝐵))
12 entr 6685 . . . 4 (((inl “ 𝐴) ≈ 𝐵𝐵 ≈ (inl “ 𝐵)) → (inl “ 𝐴) ≈ (inl “ 𝐵))
138, 11, 12syl2anc 409 . . 3 ((𝐴𝐵𝐶𝐷) → (inl “ 𝐴) ≈ (inl “ 𝐵))
14 encv 6647 . . . . . . . 8 (𝐶𝐷 → (𝐶 ∈ V ∧ 𝐷 ∈ V))
1514adantl 275 . . . . . . 7 ((𝐴𝐵𝐶𝐷) → (𝐶 ∈ V ∧ 𝐷 ∈ V))
1615simpld 111 . . . . . 6 ((𝐴𝐵𝐶𝐷) → 𝐶 ∈ V)
17 eninr 6990 . . . . . 6 (𝐶 ∈ V → (inr “ 𝐶) ≈ 𝐶)
1816, 17syl 14 . . . . 5 ((𝐴𝐵𝐶𝐷) → (inr “ 𝐶) ≈ 𝐶)
19 entr 6685 . . . . 5 (((inr “ 𝐶) ≈ 𝐶𝐶𝐷) → (inr “ 𝐶) ≈ 𝐷)
2018, 19sylancom 417 . . . 4 ((𝐴𝐵𝐶𝐷) → (inr “ 𝐶) ≈ 𝐷)
21 eninr 6990 . . . . . 6 (𝐷 ∈ V → (inr “ 𝐷) ≈ 𝐷)
2215, 21simpl2im 384 . . . . 5 ((𝐴𝐵𝐶𝐷) → (inr “ 𝐷) ≈ 𝐷)
2322ensymd 6684 . . . 4 ((𝐴𝐵𝐶𝐷) → 𝐷 ≈ (inr “ 𝐷))
24 entr 6685 . . . 4 (((inr “ 𝐶) ≈ 𝐷𝐷 ≈ (inr “ 𝐷)) → (inr “ 𝐶) ≈ (inr “ 𝐷))
2520, 23, 24syl2anc 409 . . 3 ((𝐴𝐵𝐶𝐷) → (inr “ 𝐶) ≈ (inr “ 𝐷))
26 djuin 6956 . . . 4 ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅
2726a1i 9 . . 3 ((𝐴𝐵𝐶𝐷) → ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅)
28 djuin 6956 . . . 4 ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅
2928a1i 9 . . 3 ((𝐴𝐵𝐶𝐷) → ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅)
30 unen 6717 . . 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 1218 . 2 ((𝐴𝐵𝐶𝐷) → ((inl “ 𝐴) ∪ (inr “ 𝐶)) ≈ ((inl “ 𝐵) ∪ (inr “ 𝐷)))
32 djuun 6959 . 2 ((inl “ 𝐴) ∪ (inr “ 𝐶)) = (𝐴𝐶)
33 djuun 6959 . 2 ((inl “ 𝐵) ∪ (inr “ 𝐷)) = (𝐵𝐷)
3431, 32, 333brtr3g 3968 1 ((𝐴𝐵𝐶𝐷) → (𝐴𝐶) ≈ (𝐵𝐷))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332   ∈ wcel 1481  Vcvv 2689   ∪ cun 3073   ∩ cin 3074  ∅c0 3367   class class class wbr 3936   “ cima 4549   ≈ cen 6639   ⊔ cdju 6929  inlcinl 6937  inrcinr 6938 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-coll 4050  ax-sep 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-tr 4034  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-1st 6045  df-2nd 6046  df-1o 6320  df-er 6436  df-en 6642  df-dju 6930  df-inl 6939  df-inr 6940 This theorem is referenced by:  djuenun  7084  exmidunben  11973  enctlem  11979
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