Theorem bnj1146 31743

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

Hypothesis

Ref	Expression
bnj1146.1	⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)

Assertion

Ref	Expression
bnj1146	⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵

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

Allowed substitution hint:   𝐴(𝑥)

Proof of Theorem bnj1146

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		nfv 1874	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)
2		bnj1146.1	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
3	2	nf5i 2085	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfv 1874	. . . . . . 7 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐵
5	3, 4	nfan 1863	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)
6		eleq1w 2841	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
7	6	anbi1d 621	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)))
8	1, 5, 7	cbvexv1 2279	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))
9		df-rex 3087	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))
10		df-rex 3087	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))
11	8, 9, 10	3bitr4i 295	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵)
12	11	abbii 2837	. . 3 ⊢ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵} = {𝑤 ∣ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵}
13		df-iun 4790	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵}
14		df-iun 4790	. . 3 ⊢ ∪ 𝑦 ∈ 𝐴 𝐵 = {𝑤 ∣ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵}
15	12, 13, 14	3eqtr4i 2805	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐵
16		bnj1143 31742	. 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐵 ⊆ 𝐵
17	15, 16	eqsstri 3884	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵

Syntax hints:   → wi 4   ∧ wa 387  ∀wal 1506  ∃wex 1743   ∈ wcel 2051  {cab 2751  ∃wrex 3082   ⊆ wss 3822  ∪ ciun 4788

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-v 3410  df-dif 3825  df-in 3829  df-ss 3836  df-nul 4173  df-iun 4790

This theorem is referenced by:  bnj1145  31942