Theorem iuniin 5008

Description: Law combining indexed union with indexed intersection. Eq. 14 in [KuratowskiMostowski] p. 109. This theorem also appears as the last example at http://en.wikipedia.org/wiki/Union%5F%28set%5Ftheory%29. (Contributed by NM, 17-Aug-2004.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
iuniin	⊢ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ⊆ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶

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

Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem iuniin

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		r19.12 3311	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶 → ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
2		eliin 5000	. . . . . 6 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶))
3	2	elv 3482	. . . . 5 ⊢ (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
4	3	rexbii 3091	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐵 𝑧 ∈ 𝐶)
5		eliun 4999	. . . . 5 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
6	5	ralbii 3090	. . . 4 ⊢ (∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐶)
7	1, 4, 6	3imtr4i 292	. . 3 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶 → ∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
8		eliun 4999	. . 3 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 𝐶)
9		eliin 5000	. . . 4 ⊢ (𝑧 ∈ V → (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶))
10	9	elv 3482	. . 3 ⊢ (𝑧 ∈ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶 ↔ ∀𝑦 ∈ 𝐵 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐶)
11	7, 8, 10	3imtr4i 292	. 2 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 → 𝑧 ∈ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶)
12	11	ssriv 3998	1 ⊢ ∪ 𝑥 ∈ 𝐴 ∩ 𝑦 ∈ 𝐵 𝐶 ⊆ ∩ 𝑦 ∈ 𝐵 ∪ 𝑥 ∈ 𝐴 𝐶

Syntax hints:   ↔ wb 206   ∈ wcel 2105  ∀wral 3058  ∃wrex 3067  Vcvv 3477   ⊆ wss 3962  ∪ ciun 4995  ∩ ciin 4996

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1539  df-ex 1776  df-nf 1780  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-ral 3059  df-rex 3068  df-v 3479  df-ss 3979  df-iun 4997  df-iin 4998

This theorem is referenced by: (None)