MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  euelss Structured version   Visualization version   GIF version

Theorem euelss 4057
Description: Transfer uniqueness of an element to a smaller subclass. (Contributed by AV, 14-Apr-2020.)
Assertion
Ref Expression
euelss ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴 ∧ ∃!𝑥 𝑥𝐵) → ∃!𝑥 𝑥𝐴)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem euelss
StepHypRef Expression
1 id 22 . . . 4 (𝐴𝐵𝐴𝐵)
2 df-rex 3056 . . . . 5 (∃𝑥𝐴 ⊤ ↔ ∃𝑥(𝑥𝐴 ∧ ⊤))
3 ancom 465 . . . . . . 7 ((𝑥𝐴 ∧ ⊤) ↔ (⊤ ∧ 𝑥𝐴))
4 truan 1650 . . . . . . 7 ((⊤ ∧ 𝑥𝐴) ↔ 𝑥𝐴)
53, 4bitri 264 . . . . . 6 ((𝑥𝐴 ∧ ⊤) ↔ 𝑥𝐴)
65exbii 1923 . . . . 5 (∃𝑥(𝑥𝐴 ∧ ⊤) ↔ ∃𝑥 𝑥𝐴)
72, 6sylbbr 226 . . . 4 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 ⊤)
8 df-reu 3057 . . . . 5 (∃!𝑥𝐵 ⊤ ↔ ∃!𝑥(𝑥𝐵 ∧ ⊤))
9 ancom 465 . . . . . . 7 ((𝑥𝐵 ∧ ⊤) ↔ (⊤ ∧ 𝑥𝐵))
10 truan 1650 . . . . . . 7 ((⊤ ∧ 𝑥𝐵) ↔ 𝑥𝐵)
119, 10bitri 264 . . . . . 6 ((𝑥𝐵 ∧ ⊤) ↔ 𝑥𝐵)
1211eubii 2629 . . . . 5 (∃!𝑥(𝑥𝐵 ∧ ⊤) ↔ ∃!𝑥 𝑥𝐵)
138, 12sylbbr 226 . . . 4 (∃!𝑥 𝑥𝐵 → ∃!𝑥𝐵 ⊤)
14 reuss 4051 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 ⊤ ∧ ∃!𝑥𝐵 ⊤) → ∃!𝑥𝐴 ⊤)
151, 7, 13, 14syl3an 1164 . . 3 ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴 ∧ ∃!𝑥 𝑥𝐵) → ∃!𝑥𝐴 ⊤)
16 df-reu 3057 . . 3 (∃!𝑥𝐴 ⊤ ↔ ∃!𝑥(𝑥𝐴 ∧ ⊤))
1715, 16sylib 208 . 2 ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴 ∧ ∃!𝑥 𝑥𝐵) → ∃!𝑥(𝑥𝐴 ∧ ⊤))
18 ancom 465 . . . 4 ((⊤ ∧ 𝑥𝐴) ↔ (𝑥𝐴 ∧ ⊤))
194, 18bitr3i 266 . . 3 (𝑥𝐴 ↔ (𝑥𝐴 ∧ ⊤))
2019eubii 2629 . 2 (∃!𝑥 𝑥𝐴 ↔ ∃!𝑥(𝑥𝐴 ∧ ⊤))
2117, 20sylibr 224 1 ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴 ∧ ∃!𝑥 𝑥𝐵) → ∃!𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1072  wtru 1633  wex 1853  wcel 2139  ∃!weu 2607  wrex 3051  ∃!wreu 3052  wss 3715
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-ral 3055  df-rex 3056  df-reu 3057  df-in 3722  df-ss 3729
This theorem is referenced by:  initoeu1  16882  termoeu1  16889
  Copyright terms: Public domain W3C validator