Theorem bj-uniex2 15408

Description: uniex2 4467 from bounded separation. (Contributed by BJ, 15-Oct-2019.) (Proof modification is discouraged.)

Assertion

Ref	Expression
bj-uniex2	⊢ ∃𝑦 𝑦 = ∪ 𝑥

Distinct variable group:   𝑥,𝑦

Proof of Theorem bj-uniex2

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bdcuni 15368	. . . 4 ⊢ BOUNDED ∪ 𝑥
2	1	bdeli 15338	. . 3 ⊢ BOUNDED 𝑧 ∈ ∪ 𝑥
3		zfun 4465	. . . 4 ⊢ ∃𝑦∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦)
4		eluni 3838	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑥 ↔ ∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥))
5	4	imbi1i 238	. . . . . 6 ⊢ ((𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ (∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
6	5	albii 1481	. . . . 5 ⊢ (∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
7	6	exbii 1616	. . . 4 ⊢ (∃𝑦∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦) ↔ ∃𝑦∀𝑧(∃𝑦(𝑧 ∈ 𝑦 ∧ 𝑦 ∈ 𝑥) → 𝑧 ∈ 𝑦))
8	3, 7	mpbir 146	. . 3 ⊢ ∃𝑦∀𝑧(𝑧 ∈ ∪ 𝑥 → 𝑧 ∈ 𝑦)
9	2, 8	bdbm1.3ii 15383	. 2 ⊢ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥)
10		dfcleq 2187	. . 3 ⊢ (𝑦 = ∪ 𝑥 ↔ ∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥))
11	10	exbii 1616	. 2 ⊢ (∃𝑦 𝑦 = ∪ 𝑥 ↔ ∃𝑦∀𝑧(𝑧 ∈ 𝑦 ↔ 𝑧 ∈ ∪ 𝑥))
12	9, 11	mpbir 146	1 ⊢ ∃𝑦 𝑦 = ∪ 𝑥

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   = wceq 1364  ∃wex 1503   ∈ wcel 2164  ∪ cuni 3835

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-un 4464  ax-bd0 15305  ax-bdex 15311  ax-bdel 15313  ax-bdsb 15314  ax-bdsep 15376

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-rex 2478  df-v 2762  df-uni 3836  df-bdc 15333

This theorem is referenced by:  bj-uniex  15409