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Theorem euelss 3872
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 2901 . . . . 5 (∃𝑥𝐴 ⊤ ↔ ∃𝑥(𝑥𝐴 ∧ ⊤))
3 ancom 464 . . . . . . 7 ((𝑥𝐴 ∧ ⊤) ↔ (⊤ ∧ 𝑥𝐴))
4 truan 1491 . . . . . . 7 ((⊤ ∧ 𝑥𝐴) ↔ 𝑥𝐴)
53, 4bitri 262 . . . . . 6 ((𝑥𝐴 ∧ ⊤) ↔ 𝑥𝐴)
65exbii 1763 . . . . 5 (∃𝑥(𝑥𝐴 ∧ ⊤) ↔ ∃𝑥 𝑥𝐴)
72, 6sylbbr 224 . . . 4 (∃𝑥 𝑥𝐴 → ∃𝑥𝐴 ⊤)
8 df-reu 2902 . . . . 5 (∃!𝑥𝐵 ⊤ ↔ ∃!𝑥(𝑥𝐵 ∧ ⊤))
9 ancom 464 . . . . . . 7 ((𝑥𝐵 ∧ ⊤) ↔ (⊤ ∧ 𝑥𝐵))
10 truan 1491 . . . . . . 7 ((⊤ ∧ 𝑥𝐵) ↔ 𝑥𝐵)
119, 10bitri 262 . . . . . 6 ((𝑥𝐵 ∧ ⊤) ↔ 𝑥𝐵)
1211eubii 2479 . . . . 5 (∃!𝑥(𝑥𝐵 ∧ ⊤) ↔ ∃!𝑥 𝑥𝐵)
138, 12sylbbr 224 . . . 4 (∃!𝑥 𝑥𝐵 → ∃!𝑥𝐵 ⊤)
14 reuss 3866 . . . 4 ((𝐴𝐵 ∧ ∃𝑥𝐴 ⊤ ∧ ∃!𝑥𝐵 ⊤) → ∃!𝑥𝐴 ⊤)
151, 7, 13, 14syl3an 1359 . . 3 ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴 ∧ ∃!𝑥 𝑥𝐵) → ∃!𝑥𝐴 ⊤)
16 df-reu 2902 . . 3 (∃!𝑥𝐴 ⊤ ↔ ∃!𝑥(𝑥𝐴 ∧ ⊤))
1715, 16sylib 206 . 2 ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴 ∧ ∃!𝑥 𝑥𝐵) → ∃!𝑥(𝑥𝐴 ∧ ⊤))
18 ancom 464 . . . 4 ((⊤ ∧ 𝑥𝐴) ↔ (𝑥𝐴 ∧ ⊤))
194, 18bitr3i 264 . . 3 (𝑥𝐴 ↔ (𝑥𝐴 ∧ ⊤))
2019eubii 2479 . 2 (∃!𝑥 𝑥𝐴 ↔ ∃!𝑥(𝑥𝐴 ∧ ⊤))
2117, 20sylibr 222 1 ((𝐴𝐵 ∧ ∃𝑥 𝑥𝐴 ∧ ∃!𝑥 𝑥𝐵) → ∃!𝑥 𝑥𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  w3a 1030  wtru 1475  wex 1694  wcel 1976  ∃!weu 2457  wrex 2896  ∃!wreu 2897  wss 3539
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-ral 2900  df-rex 2901  df-reu 2902  df-in 3546  df-ss 3553
This theorem is referenced by:  initoeu1  16430  termoeu1  16437
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