Theorem unen 4440

Description: Equinumerosity of union of disjoint sets. Theorem 4 of [Suppes] p. 92.

Assertion

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

Proof of Theorem unen

Step	Hyp	Ref	Expression
1		unexb 2879	. . . . 5 ⊢ ((B ∈ V ⋀ D ∈ V) ↔ (B ∪ D) ∈ V)
2		breng 4381	. . . . . 6 ⊢ (B ∈ V → (A ≈ B ↔ ∃f f:A–1-1-onto→B))
3		breng 4381	. . . . . 6 ⊢ (D ∈ V → (C ≈ D ↔ ∃g g:C–1-1-onto→D))
4	2, 3	bi2anan9 634	. . . . 5 ⊢ ((B ∈ V ⋀ D ∈ V) → ((A ≈ B ⋀ C ≈ D) ↔ (∃f f:A–1-1-onto→B ⋀ ∃g g:C–1-1-onto→D)))
5	1, 4	sylbir 201	. . . 4 ⊢ ((B ∪ D) ∈ V → ((A ≈ B ⋀ C ≈ D) ↔ (∃f f:A–1-1-onto→B ⋀ ∃g g:C–1-1-onto→D)))
6		breng 4381	. . . . . . . 8 ⊢ ((B ∪ D) ∈ V → ((A ∪ C) ≈ (B ∪ D) ↔ ∃h h:(A ∪ C)–1-1-onto→(B ∪ D)))
7		f1oun 3712	. . . . . . . . 9 ⊢ (((f:A–1-1-onto→B ⋀ g:C–1-1-onto→D) ⋀ ((A ∩ C) = ∅ ⋀ (B ∩ D) = ∅)) → (f ∪ g):(A ∪ C)–1-1-onto→(B ∪ D))
8		visset 1816	. . . . . . . . . . 11 ⊢ f ∈ V
9		visset 1816	. . . . . . . . . . 11 ⊢ g ∈ V
10	8, 9	unex 2878	. . . . . . . . . 10 ⊢ (f ∪ g) ∈ V
11		f1oeq1 3690	. . . . . . . . . 10 ⊢ (h = (f ∪ g) → (h:(A ∪ C)–1-1-onto→(B ∪ D) ↔ (f ∪ g):(A ∪ C)–1-1-onto→(B ∪ D)))
12	10, 11	cla4ev 1872	. . . . . . . . 9 ⊢ ((f ∪ g):(A ∪ C)–1-1-onto→(B ∪ D) → ∃h h:(A ∪ C)–1-1-onto→(B ∪ D))
13	7, 12	syl 10	. . . . . . . 8 ⊢ (((f:A–1-1-onto→B ⋀ g:C–1-1-onto→D) ⋀ ((A ∩ C) = ∅ ⋀ (B ∩ D) = ∅)) → ∃h h:(A ∪ C)–1-1-onto→(B ∪ D))
14	6, 13	syl5bir 210	. . . . . . 7 ⊢ ((B ∪ D) ∈ V → (((f:A–1-1-onto→B ⋀ g:C–1-1-onto→D) ⋀ ((A ∩ C) = ∅ ⋀ (B ∩ D) = ∅)) → (A ∪ C) ≈ (B ∪ D)))
15	14	exp3a 376	. . . . . 6 ⊢ ((B ∪ D) ∈ V → ((f:A–1-1-onto→B ⋀ g:C–1-1-onto→D) → (((A ∩ C) = ∅ ⋀ (B ∩ D) = ∅) → (A ∪ C) ≈ (B ∪ D))))
16	15	19.23advv 1299	. . . . 5 ⊢ ((B ∪ D) ∈ V → (∃f∃g(f:A–1-1-onto→B ⋀ g:C–1-1-onto→D) → (((A ∩ C) = ∅ ⋀ (B ∩ D) = ∅) → (A ∪ C) ≈ (B ∪ D))))
17		eeanv 1325	. . . . 5 ⊢ (∃f∃g(f:A–1-1-onto→B ⋀ g:C–1-1-onto→D) ↔ (∃f f:A–1-1-onto→B ⋀ ∃g g:C–1-1-onto→D))
18	16, 17	syl5ibr 207	. . . 4 ⊢ ((B ∪ D) ∈ V → ((∃f f:A–1-1-onto→B ⋀ ∃g g:C–1-1-onto→D) → (((A ∩ C) = ∅ ⋀ (B ∩ D) = ∅) → (A ∪ C) ≈ (B ∪ D))))
19	5, 18	sylbid 203	. . 3 ⊢ ((B ∪ D) ∈ V → ((A ≈ B ⋀ C ≈ D) → (((A ∩ C) = ∅ ⋀ (B ∩ D) = ∅) → (A ∪ C) ≈ (B ∪ D))))
20	19	imp3a 361	. 2 ⊢ ((B ∪ D) ∈ V → (((A ≈ B ⋀ C ≈ D) ⋀ ((A ∩ C) = ∅ ⋀ (B ∩ D) = ∅)) → (A ∪ C) ≈ (B ∪ D)))
21		brprc 2666	. . . 4 ⊢ (¬ (B ∪ D) ∈ V → ((A ∪ C) ≈ (B ∪ D) ↔ (A ∪ C) ≈ (A ∪ C)))
22		relen 4378	. . . . . . . 8 ⊢ Rel ≈
23	22	brrelexi 3214	. . . . . . 7 ⊢ (A ≈ B → A ∈ V)
24	22	brrelexi 3214	. . . . . . 7 ⊢ (C ≈ D → C ∈ V)
25	23, 24	anim12i 333	. . . . . 6 ⊢ ((A ≈ B ⋀ C ≈ D) → (A ∈ V ⋀ C ∈ V))
26		unexb 2879	. . . . . 6 ⊢ ((A ∈ V ⋀ C ∈ V) ↔ (A ∪ C) ∈ V)
27	25, 26	sylib 198	. . . . 5 ⊢ ((A ≈ B ⋀ C ≈ D) → (A ∪ C) ∈ V)
28		enrefg 4396	. . . . 5 ⊢ ((A ∪ C) ∈ V → (A ∪ C) ≈ (A ∪ C))
29	27, 28	syl 10	. . . 4 ⊢ ((A ≈ B ⋀ C ≈ D) → (A ∪ C) ≈ (A ∪ C))
30	21, 29	syl5bir 210	. . 3 ⊢ (¬ (B ∪ D) ∈ V → ((A ≈ B ⋀ C ≈ D) → (A ∪ C) ≈ (B ∪ D)))
31	30	adantrd 393	. 2 ⊢ (¬ (B ∪ D) ∈ V → (((A ≈ B ⋀ C ≈ D) ⋀ ((A ∩ C) = ∅ ⋀ (B ∩ D) = ∅)) → (A ∪ C) ≈ (B ∪ D)))
32	20, 31	pm2.61i 126	1 ⊢ (((A ≈ B ⋀ C ≈ D) ⋀ ((A ∩ C) = ∅ ⋀ (B ∩ D) = ∅)) → (A ∪ C) ≈ (B ∪ D))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∃wex 982  Vcvv 1814   ∪ cun 2048   ∩ cin 2049  ∅c0 2283   class class class wbr 2624  –1-1-onto→wf1o 3187   ≈ cen 4370

This theorem is referenced by:  undom 4444  limensuci 4512  phplem2 4515  pssnn 4544  unfi 4563  unfiOLD 4564  pm54.43 4581  infensuc 4648  cdaun 4934  cdaen 4936  cda1en 4938  cdacomen 4941  cdaassen 4942  xpcdaen 4943

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-en 4374

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