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Theorem snhesn 39997
Description: Any singleton is hereditary in any singleton. (Contributed by RP, 28-Mar-2020.)
Assertion
Ref Expression
snhesn {⟨𝐴, 𝐴⟩} hereditary {𝐵}

Proof of Theorem snhesn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 3503 . . . . . . 7 𝑥 ∈ V
21elima3 5934 . . . . . 6 (𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ↔ ∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}))
3 velsn 4580 . . . . . 6 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
42, 3imbi12i 352 . . . . 5 ((𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
54albii 1813 . . . 4 (∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
6 velsn 4580 . . . . . . . 8 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
7 opex 5353 . . . . . . . . . 10 𝑦, 𝑥⟩ ∈ V
87elsn 4579 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩)
9 vex 3503 . . . . . . . . . 10 𝑦 ∈ V
109, 1opth 5365 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩ ↔ (𝑦 = 𝐴𝑥 = 𝐴))
118, 10bitri 276 . . . . . . . 8 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ (𝑦 = 𝐴𝑥 = 𝐴))
126, 11anbi12i 626 . . . . . . 7 ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴𝑥 = 𝐴)))
13 3anass 1089 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴𝑥 = 𝐴)))
1412, 13bitr4i 279 . . . . . 6 ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴))
15 simp3 1132 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑥 = 𝐴)
16 simp2 1131 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑦 = 𝐴)
17 simp1 1130 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑦 = 𝐵)
1815, 16, 173eqtr2d 2867 . . . . . 6 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑥 = 𝐵)
1914, 18sylbi 218 . . . . 5 ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
2019exlimiv 1924 . . . 4 (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
215, 20mpgbir 1793 . . 3 𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵})
22 dfss2 3959 . . 3 (({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}))
2321, 22mpbir 232 . 2 ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵}
24 df-he 39984 . 2 ({⟨𝐴, 𝐴⟩} hereditary {𝐵} ↔ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵})
2523, 24mpbir 232 1 {⟨𝐴, 𝐴⟩} hereditary {𝐵}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1081  wal 1528   = wceq 1530  wex 1773  wcel 2107  wss 3940  {csn 4564  cop 4570  cima 5557   hereditary whe 39983
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-opab 5126  df-xp 5560  df-cnv 5562  df-dm 5564  df-rn 5565  df-res 5566  df-ima 5567  df-he 39984
This theorem is referenced by: (None)
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