Theorem unen 8598

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 8520	. . 3 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑥 𝑥:𝐴–1-1-onto→𝐵)
2		bren 8520	. . 3 ⊢ (𝐶 ≈ 𝐷 ↔ ∃𝑦 𝑦:𝐶–1-1-onto→𝐷)
3		exdistrv 1956	. . . 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 3499	. . . . . . . 8 ⊢ 𝑥 ∈ V
5		vex 3499	. . . . . . . 8 ⊢ 𝑦 ∈ V
6	4, 5	unex 7471	. . . . . . 7 ⊢ (𝑥 ∪ 𝑦) ∈ V
7		f1oun 6636	. . . . . . 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 8527	. . . . . . 7 ⊢ (((𝑥 ∪ 𝑦) ∈ V ∧ (𝑥 ∪ 𝑦):(𝐴 ∪ 𝐶)–1-1-onto→(𝐵 ∪ 𝐷)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))
9	6, 7, 8	sylancr 589	. . . . . 6 ⊢ (((𝑥:𝐴–1-1-onto→𝐵 ∧ 𝑦:𝐶–1-1-onto→𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))
10	9	ex 415	. . . . 5 ⊢ ((𝑥:𝐴–1-1-onto→𝐵 ∧ 𝑦:𝐶–1-1-onto→𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
11	10	exlimivv 1933	. . . 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 237	. . 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 599	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷)))
14	13	imp 409	1 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ ((𝐴 ∩ 𝐶) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝐴 ∪ 𝐶) ≈ (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 398   = wceq 1537  ∃wex 1780   ∈ wcel 2114  Vcvv 3496   ∪ cun 3936   ∩ cin 3937  ∅c0 4293   class class class wbr 5068  –1-1-onto→wf1o 6356   ≈ cen 8508

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332  ax-un 7463

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-ral 3145  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-uni 4841  df-br 5069  df-opab 5131  df-id 5462  df-xp 5563  df-rel 5564  df-cnv 5565  df-co 5566  df-dm 5567  df-rn 5568  df-fun 6359  df-fn 6360  df-f 6361  df-f1 6362  df-fo 6363  df-f1o 6364  df-en 8512

This theorem is referenced by:  enpr2d  8599  difsnen  8601  undom  8607  limensuci  8695  infensuc  8697  phplem2  8699  pssnn  8738  dif1en  8753  unfi  8787  infdifsn  9122  pm54.43  9431  dif1card  9438  endjudisj  9596  djuen  9597  ssfin4  9734  fin23lem26  9749  unsnen  9977  fzennn  13339  mreexexlem4d  16920