Theorem unen 6977

Description: Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92. (Contributed by NM, 11-Jun-1998.) (Revised by Mario Carneiro, 26-Apr-2015.)

Assertion

Ref	Expression
unen	⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))

Proof of Theorem unen

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bren 6903	. . 3 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑥 𝑥:𝐴–1-1-onto→𝐵)
2		bren 6903	. . 3 ⊢ (𝐶 ≈ 𝐷 ↔ ∃𝑦 𝑦:𝐶–1-1-onto→𝐷)
3		eeanv 1983	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥:𝐴–1-1-onto→𝐵 ∧ 𝑦:𝐶–1-1-onto→𝐷) ↔ (∃𝑥 𝑥:𝐴–1-1-onto→𝐵 ∧ ∃𝑦 𝑦:𝐶–1-1-onto→𝐷))
4		vex 2802	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 2802	. . . . . . . 8 ⊢ 𝑦 ∈ V
6	4, 5	unex 4532	. . . . . . 7 ⊢ (𝑥 ∪ 𝑦) ∈ V
7		f1oun 5594	. . . . . . 7 ⊢ (((𝑥:𝐴–1-1-onto→𝐵 ∧ 𝑦:𝐶–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝑥 ∪ 𝑦):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷))
8		f1oen3g 6913	. . . . . . 7 ⊢ (((𝑥 ∪ 𝑦) ∈ V ∧ (𝑥 ∪ 𝑦):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))
9	6, 7, 8	sylancr 414	. . . . . 6 ⊢ (((𝑥:𝐴–1-1-onto→𝐵 ∧ 𝑦:𝐶–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))
10	9	ex 115	. . . . 5 ⊢ ((𝑥:𝐴–1-1-onto→𝐵 ∧ 𝑦:𝐶–1-1-onto→𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
11	10	exlimivv 1943	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥:𝐴–1-1-onto→𝐵 ∧ 𝑦:𝐶–1-1-onto→𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
12	3, 11	sylbir 135	. . 3 ⊢ ((∃𝑥 𝑥:𝐴–1-1-onto→𝐵 ∧ ∃𝑦 𝑦:𝐶–1-1-onto→𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
13	1, 2, 12	syl2anb 291	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
14	13	imp 124	1 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799   ∪ cun 3195   ∩ cin 3196  ∅c0 3491   class class class wbr 4083  –1-1-onto→wf1o 5317   ≈ cen 6893

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-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  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-ral 2513  df-rex 2514  df-v 2801  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 3889  df-br 4084  df-opab 4146  df-id 4384  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-en 6896

This theorem is referenced by:  enpr2d  6980  phplem2  7022  fiunsnnn  7051  unsnfi  7089  endjusym  7271  pm54.43  7371  endjudisj  7400  djuen  7401  frecfzennn  10656  unennn  12976