Theorem bnj1143 30115

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 4457	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵}
2		notnotb 303	. . . . . . . 8 ⊢ (𝐴 = ∅ ↔ ¬ ¬ 𝐴 = ∅)
3		neq0 3889	. . . . . . . 8 ⊢ (¬ 𝐴 = ∅ ↔ ∃𝑥 𝑥 ∈ 𝐴)
4	2, 3	xchbinx 323	. . . . . . 7 ⊢ (𝐴 = ∅ ↔ ¬ ∃𝑥 𝑥 ∈ 𝐴)
5		df-rex 2902	. . . . . . . . 9 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵))
6		exsimpl 1783	. . . . . . . . 9 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐵) → ∃𝑥 𝑥 ∈ 𝐴)
7	5, 6	sylbi 206	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵 → ∃𝑥 𝑥 ∈ 𝐴)
8	7	con3i 149	. . . . . . 7 ⊢ (¬ ∃𝑥 𝑥 ∈ 𝐴 → ¬ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
9	4, 8	sylbi 206	. . . . . 6 ⊢ (𝐴 = ∅ → ¬ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
10	9	alrimiv 1842	. . . . 5 ⊢ (𝐴 = ∅ → ∀𝑧 ¬ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
11		notnotb 303	. . . . . . 7 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅ ↔ ¬ ¬ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅)
12		neq0 3889	. . . . . . . 8 ⊢ (¬ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ∃𝑧 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵)
13	1	eqeq1i 2615	. . . . . . . . 9 ⊢ (∪ 𝑥 ∈ 𝐴 𝐵 = ∅ ↔ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅)
14	13	notbii 309	. . . . . . . 8 ⊢ (¬ ∪ 𝑥 ∈ 𝐴 𝐵 = ∅ ↔ ¬ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅)
15		df-iun 4457	. . . . . . . . . 10 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵}
16	15	eleq2i 2680	. . . . . . . . 9 ⊢ (𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
17	16	exbii 1764	. . . . . . . 8 ⊢ (∃𝑧 𝑧 ∈ ∪ 𝑥 ∈ 𝐴 𝐵 ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
18	12, 14, 17	3bitr3i 289	. . . . . . 7 ⊢ (¬ {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅ ↔ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
19	11, 18	xchbinx 323	. . . . . 6 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅ ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
20		alnex 1697	. . . . . 6 ⊢ (∀𝑧 ¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ ¬ ∃𝑧 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵})
21		abid 2598	. . . . . . . 8 ⊢ (𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
22	21	notbii 309	. . . . . . 7 ⊢ (¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ ¬ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
23	22	albii 1737	. . . . . 6 ⊢ (∀𝑧 ¬ 𝑧 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵} ↔ ∀𝑧 ¬ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
24	19, 20, 23	3bitr2i 287	. . . . 5 ⊢ ({𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅ ↔ ∀𝑧 ¬ ∃𝑥 ∈ 𝐴 𝑧 ∈ 𝐵)
25	10, 24	sylibr 223	. . . 4 ⊢ (𝐴 = ∅ → {𝑦 ∣ ∃𝑥 ∈ 𝐴 𝑦 ∈ 𝐵} = ∅)
26	1, 25	syl5eq 2656	. . 3 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = ∅)
27		0ss 3924	. . 3 ⊢ ∅ ⊆ 𝐵
28	26, 27	syl6eqss 3618	. 2 ⊢ (𝐴 = ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵)
29		iunconst 4465	. . 3 ⊢ (𝐴 ≠ ∅ → ∪ 𝑥 ∈ 𝐴 𝐵 = 𝐵)
30		eqimss 3620	. . 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 2865	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵

Syntax hints:  ¬ wn 3   ∧ wa 383  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596   ≠ wne 2780  ∃wrex 2897   ⊆ wss 3540  ∅c0 3874  ∪ ciun 4455

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-iun 4457

This theorem is referenced by:  bnj1146  30116  bnj1145  30315  bnj1136  30319