Theorem cplem2 4713

Description: Lemma for the Collection Principle cp 4714.

Hypothesis

Ref	Expression
cplem2.1	⊢ A ∈ V

Assertion

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

Distinct variable groups:   x,y,A   y,B

Proof of Theorem cplem2

Step	Hyp	Ref	Expression
1		eqid 1473	. . 3 ⊢ {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} = {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)}
2		eqid 1473	. . 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 4712	. 2 ⊢ ∀x ∈ A (B ≠ ∅ → (B ∩ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)}) ≠ ∅)
4		cplem2.1	. . . 4 ⊢ A ∈ V
5		scottex 4708	. . . 4 ⊢ {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} ∈ V
6	4, 5	iunex 3865	. . 3 ⊢ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} ∈ V
7		hbiu1 2580	. . . . 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 1564	. . . 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 2207	. . . . . 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)}))
10	9	neeq1d 1591	. . . . 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)}) ≠ ∅))
11	10	imbi2d 611	. . . 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)}) ≠ ∅)))
12	8, 11	ralbid 1658	. . 3 ⊢ (y = ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)} → (∀x ∈ A (B ≠ ∅ → (B ∩ y) ≠ ∅) ↔ ∀x ∈ A (B ≠ ∅ → (B ∩ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)}) ≠ ∅)))
13	6, 12	cla4ev 1865	. 2 ⊢ (∀x ∈ A (B ≠ ∅ → (B ∩ ∪x ∈ A {z ∈ B∣∀w ∈ B (rank ‘z) ⊆ (rank ‘w)}) ≠ ∅) → ∃y∀x ∈ A (B ≠ ∅ → (B ∩ y) ≠ ∅))
14	3, 13	ax-mp 7	1 ⊢ ∃y∀x ∈ A (B ≠ ∅ → (B ∩ y) ≠ ∅)

Syntax hints:   → wi 3   = wceq 954   ∈ wcel 956  ∃wex 978   ≠ wne 1582  ∀wral 1642  {crab 1645  Vcvv 1807   ∩ cin 2042   ⊆ wss 2043  ∅c0 2276  ∪ciun 2562   ‘cfv 3182  rankcrnk 4634

This theorem is referenced by:  cp 4714

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-rep 2689  ax-sep 2699  ax-nul 2706  ax-pow 2738  ax-pr 2775  ax-un 2865  ax-reg 4585  ax-inf2 4617

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 775  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-rab 1649  df-v 1808  df-sbc 1938  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-if 2358  df-pw 2398  df-sn 2408  df-pr 2409  df-tp 2411  df-op 2412  df-uni 2500  df-int 2530  df-iun 2564  df-iin 2565  df-br 2616  df-opab 2663  df-tr 2677  df-eprel 2829  df-id 2832  df-po 2839  df-so 2849  df-fr 2916  df-we 2933  df-ord 2950  df-on 2951  df-lim 2952  df-suc 2953  df-om 3132  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fn 3193  df-fv 3198  df-rdg 3934  df-r1 4635  df-rank 4636

Copyright terms: Public domain