Theorem fiiu2 10377

Description: If A is the intersection of a finite set of elements of B then A ⊆ ∪B.

Assertion

Ref	Expression
fiiu2	⊢ (B ∈ C → (A ∈ (fi ‘B) → A ⊆ ∪B))

Proof of Theorem fiiu2

Step	Hyp	Ref	Expression
1		nvel 2704	. . . . 5 ⊢ ¬ V ∈ (fi ‘B)
2		eleq1 1526	. . . . . 6 ⊢ (A = V → (A ∈ (fi ‘B) ↔ V ∈ (fi ‘B)))
3	2	biimpcd 155	. . . . 5 ⊢ (A ∈ (fi ‘B) → (A = V → V ∈ (fi ‘B)))
4	1, 3	mtoi 107	. . . 4 ⊢ (A ∈ (fi ‘B) → ¬ A = V)
5		sppfi 10376	. . . . . . . 8 ⊢ ((A ∈ (fi ‘B) ⋀ B ∈ C) → (A ∈ (fi ‘B) ↔ ∃y(y ⊆ B ⋀ ∃z ∈ ω y ≈ z ⋀ A = ∩y)))
6		eqeq1 1473	. . . . . . . . . . . . . . . 16 ⊢ (A = ∩y → (A = V ↔ ∩y = V))
7	6	negbid 609	. . . . . . . . . . . . . . 15 ⊢ (A = ∩y → (¬ A = V ↔ ¬ ∩y = V))
8		int0 2537	. . . . . . . . . . . . . . . 16 ⊢ ∩∅ = V
9		neeq2 1583	. . . . . . . . . . . . . . . . . 18 ⊢ (∩∅ = V → (∩y ≠ ∩∅ ↔ ∩y ≠ V))
10		inteq 2526	. . . . . . . . . . . . . . . . . . 19 ⊢ (y = ∅ → ∩y = ∩∅)
11	10	necon3i 1597	. . . . . . . . . . . . . . . . . 18 ⊢ (∩y ≠ ∩∅ → y ≠ ∅)
12	9, 11	syl6bir 215	. . . . . . . . . . . . . . . . 17 ⊢ (∩∅ = V → (∩y ≠ V → y ≠ ∅))
13		df-ne 1579	. . . . . . . . . . . . . . . . 17 ⊢ (∩y ≠ V ↔ ¬ ∩y = V)
14	12, 13	syl5ibr 207	. . . . . . . . . . . . . . . 16 ⊢ (∩∅ = V → (¬ ∩y = V → y ≠ ∅))
15	8, 14	ax-mp 7	. . . . . . . . . . . . . . 15 ⊢ (¬ ∩y = V → y ≠ ∅)
16	7, 15	syl6bi 214	. . . . . . . . . . . . . 14 ⊢ (A = ∩y → (¬ A = V → y ≠ ∅))
17		sseq1 2072	. . . . . . . . . . . . . . . . . 18 ⊢ (∩y = A → (∩y ⊆ ∪B ↔ A ⊆ ∪B))
18	17	biimpd 153	. . . . . . . . . . . . . . . . 17 ⊢ (∩y = A → (∩y ⊆ ∪B → A ⊆ ∪B))
19	18	eqcoms 1470	. . . . . . . . . . . . . . . 16 ⊢ (A = ∩y → (∩y ⊆ ∪B → A ⊆ ∪B))
20		intssuni2 2546	. . . . . . . . . . . . . . . 16 ⊢ ((y ⊆ B ⋀ y ≠ ∅) → ∩y ⊆ ∪B)
21	19, 20	syl5com 52	. . . . . . . . . . . . . . 15 ⊢ ((y ⊆ B ⋀ y ≠ ∅) → (A = ∩y → A ⊆ ∪B))
22	21	expcom 374	. . . . . . . . . . . . . 14 ⊢ (y ≠ ∅ → (y ⊆ B → (A = ∩y → A ⊆ ∪B)))
23	16, 22	syl6 22	. . . . . . . . . . . . 13 ⊢ (A = ∩y → (¬ A = V → (y ⊆ B → (A = ∩y → A ⊆ ∪B))))
24	23	com24 37	. . . . . . . . . . . 12 ⊢ (A = ∩y → (A = ∩y → (y ⊆ B → (¬ A = V → A ⊆ ∪B))))
25	24	pm2.43i 64	. . . . . . . . . . 11 ⊢ (A = ∩y → (y ⊆ B → (¬ A = V → A ⊆ ∪B)))
26	25	impcom 351	. . . . . . . . . 10 ⊢ ((y ⊆ B ⋀ A = ∩y) → (¬ A = V → A ⊆ ∪B))
27	26	3adant2 796	. . . . . . . . 9 ⊢ ((y ⊆ B ⋀ ∃z ∈ ω y ≈ z ⋀ A = ∩y) → (¬ A = V → A ⊆ ∪B))
28	27	19.23aiv 1290	. . . . . . . 8 ⊢ (∃y(y ⊆ B ⋀ ∃z ∈ ω y ≈ z ⋀ A = ∩y) → (¬ A = V → A ⊆ ∪B))
29	5, 28	syl6bi 214	. . . . . . 7 ⊢ ((A ∈ (fi ‘B) ⋀ B ∈ C) → (A ∈ (fi ‘B) → (¬ A = V → A ⊆ ∪B)))
30	29	ex 373	. . . . . 6 ⊢ (A ∈ (fi ‘B) → (B ∈ C → (A ∈ (fi ‘B) → (¬ A = V → A ⊆ ∪B))))
31	30	com23 32	. . . . 5 ⊢ (A ∈ (fi ‘B) → (A ∈ (fi ‘B) → (B ∈ C → (¬ A = V → A ⊆ ∪B))))
32	31	com34 36	. . . 4 ⊢ (A ∈ (fi ‘B) → (A ∈ (fi ‘B) → (¬ A = V → (B ∈ C → A ⊆ ∪B))))
33	4, 32	mpid 47	. . 3 ⊢ (A ∈ (fi ‘B) → (A ∈ (fi ‘B) → (B ∈ C → A ⊆ ∪B)))
34	33	pm2.43i 64	. 2 ⊢ (A ∈ (fi ‘B) → (B ∈ C → A ⊆ ∪B))
35	34	com12 11	1 ⊢ (B ∈ C → (A ∈ (fi ‘B) → A ⊆ ∪B))

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223   ⋀ w3a 773   = wceq 953   ∈ wcel 955  ∃wex 977   ≠ wne 1577  ∃wrex 1638  Vcvv 1802   ⊆ wss 2037  ∅c0 2270  ∪cuni 2493  ∩cint 2523   class class class wbr 2609  ωcom 3121   ‘cfv 3172   ≈ cen 4348  ficfi 10372

This theorem is referenced by:  fgsb2 10449

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-9 962  ax-10 963  ax-11 964  ax-12 965  ax-13 966  ax-14 967  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452  ax-sep 2693  ax-pow 2732  ax-pr 2769  ax-un 2857

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 775  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-ral 1641  df-rex 1642  df-rab 1644  df-v 1803  df-dif 2039  df-un 2040  df-in 2041  df-ss 2043  df-nul 2271  df-pw 2392  df-sn 2402  df-pr 2403  df-op 2406  df-uni 2494  df-int 2524  df-br 2610  df-opab 2657  df-id 2824  df-xp 3174  df-rel 3175  df-cnv 3176  df-co 3177  df-dm 3178  df-rn 3179  df-res 3180  df-ima 3181  df-fun 3182  df-fv 3188  df-fiNEW 10373

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