Theorem unen 9104

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 9014	. . 3 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑥 𝑥:𝐴⟶1-1onto𝐵)
2		bren 9014	. . 3 ⊢ (𝐶 ≈ 𝐷 ↔ ∃𝑦 𝑦:𝐶⟶1-1onto𝐷)
3		exdistrv 1990	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥:𝐴⟶1-1onto𝐵 ∧ 𝑦:𝐶⟶1-1onto𝐷) ↔ (∃𝑥 𝑥:𝐴⟶1-1onto𝐵 ∧ ∃𝑦 𝑦:𝐶⟶1-1onto𝐷))
4		vex 3458	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 3458	. . . . . . . 8 ⊢ 𝑦 ∈ V
6	4, 5	unex 7788	. . . . . . 7 ⊢ (𝑥 ∪ 𝑦) ∈ V
7		f1oun 6869	. . . . . . 7 ⊢ (((𝑥:𝐴⟶1-1onto𝐵 ∧ 𝑦:𝐶⟶1-1onto𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝑥 ∪ 𝑦):(𝐴 ∪ 𝐶)⟶1-1onto(𝐵 ∪ 𝐷))
8		f1oen3g 9024	. . . . . . 7 ⊢ (((𝑥 ∪ 𝑦) ∈ V ∧ (𝑥 ∪ 𝑦):(𝐴 ∪ 𝐶)⟶1-1onto(𝐵 ∪ 𝐷)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))
9	6, 7, 8	sylancr 599	. . . . . 6 ⊢ (((𝑥:𝐴⟶1-1onto𝐵 ∧ 𝑦:𝐶⟶1-1onto𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))
10	9	ex 418	. . . . 5 ⊢ ((𝑥:𝐴⟶1-1onto𝐵 ∧ 𝑦:𝐶⟶1-1onto𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
11	10	exlimivv 1967	. . . 4 ⊢ (∃𝑥∃𝑦(𝑥:𝐴⟶1-1onto𝐵 ∧ 𝑦:𝐶⟶1-1onto𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
12	3, 11	sylbir 238	. . 3 ⊢ ((∃𝑥 𝑥:𝐴⟶1-1onto𝐵 ∧ ∃𝑦 𝑦:𝐶⟶1-1onto𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
13	1, 2, 12	syl2anb 610	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
14	13	imp 412	1 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ∩ cin 4092   ∪ cun 4152  ∅c0 4281   class class class wbr 5106  ⟶1-1ontowf1o 6560   ≈ cen 9001

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396  ax-un 7778

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-en 9005

This theorem is used by:  enrefnn  9105  difsnen  9109  limensuci  9203  infensuc  9205  pssnn  9215  unfi  9217  dif1ennnALT  9299  infdifsn  9688  pm54.43  10112  dif1card  10119  endjudisj  10277  djuen  10278  ssfin4  10418  fin23lem26  10433  unsnen  10667  fzennn  14141  mreexexlem4d  17853