Theorem bnj1143 34804

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Assertion

Ref	Expression
bnj1143	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem bnj1143

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iun 4993	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2		notnotb 315	. . . . . . . 8 ⊢ (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
3		neq0 4352	. . . . . . . 8 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
4	2, 3	xchbinx 334	. . . . . . 7 ⊢ (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴)
5		df-rex 3071	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
6		exsimpl 1868	. . . . . . . . 9 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∃𝑥 𝑥 ∈ 𝐴)
7	5, 6	sylbi 217	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐴)
8	7	con3i 154	. . . . . . 7 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 → ¬ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
9	4, 8	sylbi 217	. . . . . 6 ⊢ (𝐴 = ∅ → ¬ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
10	9	alrimiv 1927	. . . . 5 ⊢ (𝐴 = ∅ → ∀𝑧 ¬ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
11		notnotb 315	. . . . . . 7 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅ ↔ ¬ ¬ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅)
12		neq0 4352	. . . . . . . 8 ⊢ (¬ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ∃𝑧 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
13	1	eqeq1i 2742	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 = ∅ ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅)
14	13	notbii 320	. . . . . . . 8 ⊢ (¬ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ¬ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅)
15		df-iun 4993	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
16	15	eleq2i 2833	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
17	16	exbii 1848	. . . . . . . 8 ⊢ (∃𝑧 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
18	12, 14, 17	3bitr3i 301	. . . . . . 7 ⊢ (¬ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅ ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
19	11, 18	xchbinx 334	. . . . . 6 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅ ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
20		alnex 1781	. . . . . 6 ⊢ (∀𝑧 ¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
21		abid 2718	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
22	21	notbii 320	. . . . . . 7 ⊢ (¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ ¬ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
23	22	albii 1819	. . . . . 6 ⊢ (∀𝑧 ¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ ∀𝑧 ¬ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
24	19, 20, 23	3bitr2i 299	. . . . 5 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅ ↔ ∀𝑧 ¬ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
25	10, 24	sylibr 234	. . . 4 ⊢ (𝐴 = ∅ → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅)
26	1, 25	eqtrid 2789	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = ∅)
27		0ss 4400	. . 3 ⊢ ∅ ⊆ 𝐵
28	26, 27	eqsstrdi 4028	. 2 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵)
29		iunconst 5001	. . 3 ⊢ (𝐴 ≠ ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵)
30		eqimss 4042	. . 3 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵 → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵)
31	29, 30	syl 17	. 2 ⊢ (𝐴 ≠ ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵)
32	28, 31	pm2.61ine 3025	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵

Syntax hints:  ¬ wn 3   ∧ wa 395  ∀wal 1538   = wceq 1540  ∃wex 1779   ∈ wcel 2108  {cab 2714   ≠ wne 2940  ∃wrex 3070   ⊆ wss 3951  ∅c0 4333  ∪ ciun 4991

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-12 2177  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-v 3482  df-dif 3954  df-ss 3968  df-nul 4334  df-iun 4993

This theorem is referenced by:  bnj1146  34805  bnj1145  35007  bnj1136  35011