Theorem euelss 4293

Description: Transfer uniqueness of an element to a smaller subclass. (Contributed by AV, 14-Apr-2020.)

Assertion

Ref	Expression
euelss	⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴 ∧ ∃!𝑥 𝑥 ∈ 𝐵) → ∃!𝑥 𝑥 ∈ 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem euelss

Step	Hyp	Ref	Expression
1		id 23	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → 𝐴 ⊆ 𝐵)
2		df-rex 3096	. . . . 5 ⊢ (∃𝑥 ∈ 𝐴 ⊤ ↔ ∃𝑥(𝑥 ∈ 𝐴 ∧ ⊤))
3		ancom 465	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐴 ∧ ⊤) ↔ (⊤ ∧ 𝑥 ∈ 𝐴))
4		truan 1578	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) ↔ 𝑥 ∈ 𝐴)
5	3, 4	bitri 278	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ ⊤) ↔ 𝑥 ∈ 𝐴)
6	5	exbii 1875	. . . . 5 ⊢ (∃𝑥(𝑥 ∈ 𝐴 ∧ ⊤) ↔ ∃𝑥 𝑥 ∈ 𝐴)
7	2, 6	sylbbr 239	. . . 4 ⊢ (∃𝑥 𝑥 ∈ 𝐴 → ∃𝑥 ∈ 𝐴 ⊤)
8		df-reu 3377	. . . . 5 ⊢ (∃!𝑥 ∈ 𝐵 ⊤ ↔ ∃!𝑥(𝑥 ∈ 𝐵 ∧ ⊤))
9		ancom 465	. . . . . . 7 ⊢ ((𝑥 ∈ 𝐵 ∧ ⊤) ↔ (⊤ ∧ 𝑥 ∈ 𝐵))
10		truan 1578	. . . . . . 7 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐵)
11	9, 10	bitri 278	. . . . . 6 ⊢ ((𝑥 ∈ 𝐵 ∧ ⊤) ↔ 𝑥 ∈ 𝐵)
12	11	eubii 2619	. . . . 5 ⊢ (∃!𝑥(𝑥 ∈ 𝐵 ∧ ⊤) ↔ ∃!𝑥 𝑥 ∈ 𝐵)
13	8, 12	sylbbr 239	. . . 4 ⊢ (∃!𝑥 𝑥 ∈ 𝐵 → ∃!𝑥 ∈ 𝐵 ⊤)
14		reuss 4288	. . . 4 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 ∈ 𝐴 ⊤ ∧ ∃!𝑥 ∈ 𝐵 ⊤) → ∃!𝑥 ∈ 𝐴 ⊤)
15	1, 7, 13, 14	syl3an 1176	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴 ∧ ∃!𝑥 𝑥 ∈ 𝐵) → ∃!𝑥 ∈ 𝐴 ⊤)
16		df-reu 3377	. . 3 ⊢ (∃!𝑥 ∈ 𝐴 ⊤ ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ ⊤))
17	15, 16	sylib 221	. 2 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴 ∧ ∃!𝑥 𝑥 ∈ 𝐵) → ∃!𝑥(𝑥 ∈ 𝐴 ∧ ⊤))
18		ancom 465	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
19	4, 18	bitr3i 280	. . 3 ⊢ (𝑥 ∈ 𝐴 ↔ (𝑥 ∈ 𝐴 ∧ ⊤))
20	19	eubii 2619	. 2 ⊢ (∃!𝑥 𝑥 ∈ 𝐴 ↔ ∃!𝑥(𝑥 ∈ 𝐴 ∧ ⊤))
21	17, 20	sylibr 237	1 ⊢ ((𝐴 ⊆ 𝐵 ∧ ∃𝑥 𝑥 ∈ 𝐴 ∧ ∃!𝑥 𝑥 ∈ 𝐵) → ∃!𝑥 𝑥 ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1101  ⊤wtru 1568  ∃wex 1806   ∈ wcel 2149  ∃!weu 2602  ∃wrex 3095  ∃!wreu 3374   ⊆ wss 3913

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

This theorem depends on definitions:  df-bi 210  df-an 401  df-3an 1103  df-tru 1570  df-ex 1807  df-mo 2573  df-eu 2603  df-clel 2844  df-ral 3086  df-rex 3096  df-reu 3377  df-ss 3930

This theorem is referenced by:  initoeu1  18067  termoeu1  18074