Theorem bnj1146 35124

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 1941	. . . . . 6 ⊢ Ⅎ𝑦(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)
2		bnj1146.1	. . . . . . . 8 ⊢ (𝑦 ∈ 𝐴 → ∀𝑥 𝑦 ∈ 𝐴)
3	2	nf5i 2187	. . . . . . 7 ⊢ Ⅎ𝑥 𝑦 ∈ 𝐴
4		nfv 1941	. . . . . . 7 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐵
5	3, 4	nfan 1926	. . . . . 6 ⊢ Ⅎ𝑥(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)
6		eleq1w 2852	. . . . . . 7 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
7	6	anbi1d 642	. . . . . 6 ⊢ (𝑥 = 𝑦 → ((𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ↔ (𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵)))
8	1, 5, 7	cbvexv1 2380	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵) ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))
9		df-rex 3096	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))
10		df-rex 3096	. . . . 5 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑦(𝑦 ∈ 𝐴 ∧ 𝑤 ∈ 𝐵))
11	8, 9, 10	3bitr4i 306	. . . 4 ⊢ (∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵)
12	11	abbii 2836	. . 3 ⊢ {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵} = {𝑤 ∣ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵}
13		df-iun 4962	. . 3 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = {𝑤 ∣ ∃𝑥 ∈ 𝐴 𝑤 ∈ 𝐵}
14		df-iun 4962	. . 3 ⊢ ∪ 𝑦 ∈ 𝐴 𝐵 = {𝑤 ∣ ∃𝑦 ∈ 𝐴 𝑤 ∈ 𝐵}
15	12, 13, 14	3eqtr4i 2802	. 2 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 = ∪ 𝑦 ∈ 𝐴 𝐵
16		bnj1143 35123	. 2 ⊢ ∪ 𝑦 ∈ 𝐴 𝐵 ⊆ 𝐵
17	15, 16	eqsstri 3991	1 ⊢ ∪ 𝑥 ∈ 𝐴 𝐵 ⊆ 𝐵

Syntax hints:   → wi 4   ∧ wa 400  ∀wal 1565  ∃wex 1806   ∈ wcel 2149  {cab 2747  ∃wrex 3095   ⊆ wss 3913  ∪ ciun 4960

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-clab 2748  df-cleq 2761  df-clel 2844  df-ne 2965  df-ral 3086  df-rex 3096  df-v 3465  df-dif 3916  df-ss 3930  df-nul 4295  df-iun 4962

This theorem is referenced by:  bnj1145  35326