Theorem djuen 7389

Description: Disjoint unions of equinumerous sets are equinumerous. (Contributed by Jim Kingdon, 30-Jul-2023.)

Assertion

Ref	Expression
djuen	⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷))

Proof of Theorem djuen

Step	Hyp	Ref	Expression
1		encv 6891	. . . . . . . 8 ⊢ (𝐴 ≈ 𝐵 → (𝐴 ∈ V ∧ 𝐵 ∈ V))
2	1	adantr 276	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
3	2	simpld 112	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐴 ∈ V)
4		eninl 7260	. . . . . 6 ⊢ (𝐴 ∈ V → (inl “ 𝐴) ≈ 𝐴)
5	3, 4	syl 14	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inl “ 𝐴) ≈ 𝐴)
6		simpl 109	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐴 ≈ 𝐵)
7		entr 6934	. . . . 5 ⊢ (((inl “ 𝐴) ≈ 𝐴 ∧ 𝐴 ≈ 𝐵) → (inl “ 𝐴) ≈ 𝐵)
8	5, 6, 7	syl2anc 411	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inl “ 𝐴) ≈ 𝐵)
9		eninl 7260	. . . . . 6 ⊢ (𝐵 ∈ V → (inl “ 𝐵) ≈ 𝐵)
10	2, 9	simpl2im 386	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inl “ 𝐵) ≈ 𝐵)
11	10	ensymd 6933	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐵 ≈ (inl “ 𝐵))
12		entr 6934	. . . 4 ⊢ (((inl “ 𝐴) ≈ 𝐵 ∧ 𝐵 ≈ (inl “ 𝐵)) → (inl “ 𝐴) ≈ (inl “ 𝐵))
13	8, 11, 12	syl2anc 411	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inl “ 𝐴) ≈ (inl “ 𝐵))
14		encv 6891	. . . . . . . 8 ⊢ (𝐶 ≈ 𝐷 → (𝐶 ∈ V ∧ 𝐷 ∈ V))
15	14	adantl 277	. . . . . . 7 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐶 ∈ V ∧ 𝐷 ∈ V))
16	15	simpld 112	. . . . . 6 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐶 ∈ V)
17		eninr 7261	. . . . . 6 ⊢ (𝐶 ∈ V → (inr “ 𝐶) ≈ 𝐶)
18	16, 17	syl 14	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐶)
19		entr 6934	. . . . 5 ⊢ (((inr “ 𝐶) ≈ 𝐶 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐷)
20	18, 19	sylancom 420	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ 𝐷)
21		eninr 7261	. . . . . 6 ⊢ (𝐷 ∈ V → (inr “ 𝐷) ≈ 𝐷)
22	15, 21	simpl2im 386	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐷) ≈ 𝐷)
23	22	ensymd 6933	. . . 4 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → 𝐷 ≈ (inr “ 𝐷))
24		entr 6934	. . . 4 ⊢ (((inr “ 𝐶) ≈ 𝐷 ∧ 𝐷 ≈ (inr “ 𝐷)) → (inr “ 𝐶) ≈ (inr “ 𝐷))
25	20, 23, 24	syl2anc 411	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (inr “ 𝐶) ≈ (inr “ 𝐷))
26		djuin 7227	. . . 4 ⊢ ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅
27	26	a1i 9	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐴) ∩ (inr “ 𝐶)) = ∅)
28		djuin 7227	. . . 4 ⊢ ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅
29	28	a1i 9	. . 3 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐵) ∩ (inr “ 𝐷)) = ∅)
30		unen 6967	. . 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)))
31	13, 25, 27, 29, 30	syl22anc 1272	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ((inl “ 𝐴) ∪ (inr “ 𝐶)) ≈ ((inl “ 𝐵) ∪ (inr “ 𝐷)))
32		djuun 7230	. 2 ⊢ ((inl “ 𝐴) ∪ (inr “ 𝐶)) = (𝐴 ⊔ 𝐶)
33		djuun 7230	. 2 ⊢ ((inl “ 𝐵) ∪ (inr “ 𝐷)) = (𝐵 ⊔ 𝐷)
34	31, 32, 33	3brtr3g 4115	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 ⊔ 𝐶) ≈ (𝐵 ⊔ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2799   ∪ cun 3195   ∩ cin 3196  ∅c0 3491   class class class wbr 4082   “ cima 4721   ≈ cen 6883   ⊔ cdju 7200  inlcinl 7208  inrcinr 7209

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-nul 4209  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-tr 4182  df-id 4383  df-iord 4456  df-on 4458  df-suc 4461  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-1st 6284  df-2nd 6285  df-1o 6560  df-er 6678  df-en 6886  df-dju 7201  df-inl 7210  df-inr 7211

This theorem is referenced by:  djuenun  7390  exmidunben  12992  enctlem  12998