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Theorem snhesn 37562
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 3189 . . . . . . 7 𝑥 ∈ V
21elima3 5432 . . . . . 6 (𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ↔ ∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}))
3 velsn 4164 . . . . . 6 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
42, 3imbi12i 340 . . . . 5 ((𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
54albii 1744 . . . 4 (∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
6 velsn 4164 . . . . . . . 8 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
7 opex 4893 . . . . . . . . . 10 𝑦, 𝑥⟩ ∈ V
87elsn 4163 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩)
9 vex 3189 . . . . . . . . . 10 𝑦 ∈ V
109, 1opth 4905 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩ ↔ (𝑦 = 𝐴𝑥 = 𝐴))
118, 10bitri 264 . . . . . . . 8 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ (𝑦 = 𝐴𝑥 = 𝐴))
126, 11anbi12i 732 . . . . . . 7 ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴𝑥 = 𝐴)))
13 3anass 1040 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴𝑥 = 𝐴)))
1412, 13bitr4i 267 . . . . . 6 ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴))
15 simp3 1061 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑥 = 𝐴)
16 simp2 1060 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑦 = 𝐴)
17 simp1 1059 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑦 = 𝐵)
1815, 16, 173eqtr2d 2661 . . . . . 6 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑥 = 𝐵)
1914, 18sylbi 207 . . . . 5 ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
2019exlimiv 1855 . . . 4 (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
215, 20mpgbir 1723 . . 3 𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵})
22 dfss2 3572 . . 3 (({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}))
2321, 22mpbir 221 . 2 ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵}
24 df-he 37549 . 2 ({⟨𝐴, 𝐴⟩} hereditary {𝐵} ↔ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵})
2523, 24mpbir 221 1 {⟨𝐴, 𝐴⟩} hereditary {𝐵}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036  wal 1478   = wceq 1480  wex 1701  wcel 1987  wss 3555  {csn 4148  cop 4154  cima 5077   hereditary whe 37548
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-xp 5080  df-cnv 5082  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-he 37549
This theorem is referenced by: (None)
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