Theorem cplem2 3546

Description: Lemma for the Collection Principle cp 3547.

Hypothesis

Ref	Expression
cplem2.1	⊢ A ∈ V

Assertion

Ref	Expression
cplem2	⊢ ∃y∀x ∈ A (¬ B = ∅ → ¬ (B ∩ y) = ∅)

Distinct variable group(s):   x,y,A   y,B

Proof of Theorem cplem2

Step	Hyp	Ref	Expression
Z		cleqid 1102	. . 3 ⊢ {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} = {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)}
2		cleqid 1102	. . 3 ⊢ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} = ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)}
3	1, 2	cplem1 3545	. 2 ⊢ ∀x ∈ A (¬ B = ∅ → ¬ (B ∩ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)}) = ∅)
4		cplem2.1	. . . 4 ⊢ A ∈ V
5		scottex 3541	. . . 4 ⊢ {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} ∈ V
6	4, 5	iunex 2914	. . 3 ⊢ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} ∈ V
7		hbiu1 2012	. . . . 5 ⊢ (y ∈ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} → ∀x y ∈ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)})
8	7	hbeleq 1173	. . . 4 ⊢ (y = ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} → ∀x y = ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)})
9		ineq2 1639	. . . . . . 7 ⊢ (y = ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} → (B ∩ y) = (B ∩ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)}))
10	9	cleq1d 1109	. . . . . 6 ⊢ (y = ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} → ((B ∩ y) = ∅ ↔ (B ∩ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)}) = ∅))
11	10	negbid 463	. . . . 5 ⊢ (y = ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} → (¬ (B ∩ y) = ∅ ↔ ¬ (B ∩ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)}) = ∅))
12	11	imbi2d 464	. . . 4 ⊢ (y = ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} → ((¬ B = ∅ → ¬ (B ∩ y) = ∅) ↔ (¬ B = ∅ → ¬ (B ∩ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)}) = ∅)))
13	8, 12	birald 1217	. . 3 ⊢ (y = ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} → (∀x ∈ A (¬ B = ∅ → ¬ (B ∩ y) = ∅) ↔ &forll;x ∈ A (¬ B = ∅ → ¬ (B ∩ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)}) = ∅)))
14	6, 13	cla4ev 1401	. 2 ⊢ (∀x ∈ A (¬ B = ∅ → ¬ (B ∩ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)}) = ∅) → ∃y∀x ∈ A (¬ B = ∅ → ¬ (B ∩ y) = ∅))
15	3, 14	ax-mp 6	1 ⊢ ∃y∀x ∈ A (¬ B = ∅ → ¬ (B ∩ y) = ∅)

Syntax hints:  ¬ wn 1   → wi 2  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∀wral 1201  {crab 1204  Vcvv 1348   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  ∪ciun 1994   ‘cfv 2422  rankcrnk 3486

This theorem is referenced by:  cp 3547

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-un 1076  ax-pow 1077  ax-reg 1078  ax-inf 1079

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-eu 1009  df-mo 1010  df-clab 1093  df-cleq 1097  df-clel 1099  df-ne 1192  df-ral 1205  df-rex 1206  df-rab 1208  df-v 1349  df-sbc.nbsp;1441  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-pss 1494  df-nul 1708  df-if 1777  df-pw 1799  df-sn 1811  df-pr 1812  df-tp 1814  df-op 1815  df-uni 1920  df-int 1966  df-iun 1996  df-iin 1997  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-id 2125  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202  df-on 2203  df-lim 2204  df-suc 2205  df-om 2373  df-xp 2424  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fv 2438  df-rdg 2970  df-r1 3487  df-rank 3488

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