Theorem unen 9060

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 8969	. . 3 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑥 𝑥:𝐴–1-1-onto→𝐵)
2		bren 8969	. . 3 ⊢ (𝐶 ≈ 𝐷 ↔ ∃𝑦 𝑦:𝐶–1-1-onto→𝐷)
3		exdistrv 1955	. . . 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 3463	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 3463	. . . . . . . 8 ⊢ 𝑦 ∈ V
6	4, 5	unex 7738	. . . . . . 7 ⊢ (𝑥 ∪ 𝑦) ∈ V
7		f1oun 6837	. . . . . . 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 8981	. . . . . . 7 ⊢ (((𝑥 ∪ 𝑦) ∈ V ∧ (𝑥 ∪ 𝑦):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))
9	6, 7, 8	sylancr 587	. . . . . 6 ⊢ (((𝑥:𝐴–1-1-onto→𝐵 ∧ 𝑦:𝐶–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))
10	9	ex 412	. . . . 5 ⊢ ((𝑥:𝐴–1-1-onto→𝐵 ∧ 𝑦:𝐶–1-1-onto→𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
11	10	exlimivv 1932	. . . 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 235	. . 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 598	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
14	13	imp 406	1 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2108  Vcvv 3459   ∪ cun 3924   ∩ cin 3925  ∅c0 4308   class class class wbr 5119  –1-1-onto→wf1o 6530   ≈ cen 8956

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-en 8960

This theorem is referenced by:  enrefnn  9061  enpr2dOLD  9064  difsnen  9067  undomOLD  9074  limensuci  9167  infensuc  9169  pssnn  9182  unfi  9185  phplem2OLD  9229  dif1ennnALT  9283  infdifsn  9671  pm54.43  10015  dif1card  10024  endjudisj  10183  djuen  10184  ssfin4  10324  fin23lem26  10339  unsnen  10567  fzennn  13986  mreexexlem4d  17659