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Theorem snhesn 43453
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 3466 . . . . . . 7 𝑥 ∈ V
21elima3 6076 . . . . . 6 (𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ↔ ∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}))
3 velsn 4649 . . . . . 6 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
42, 3imbi12i 349 . . . . 5 ((𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
54albii 1814 . . . 4 (∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
6 velsn 4649 . . . . . . . 8 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
7 opex 5470 . . . . . . . . . 10 𝑦, 𝑥⟩ ∈ V
87elsn 4648 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩)
9 vex 3466 . . . . . . . . . 10 𝑦 ∈ V
109, 1opth 5482 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩ ↔ (𝑦 = 𝐴𝑥 = 𝐴))
118, 10bitri 274 . . . . . . . 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 1092 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴𝑥 = 𝐴)))
1412, 13bitr4i 277 . . . . . 6 ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴))
15 simp3 1135 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑥 = 𝐴)
16 simp2 1134 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑦 = 𝐴)
17 simp1 1133 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑦 = 𝐵)
1815, 16, 173eqtr2d 2772 . . . . . 6 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑥 = 𝐵)
1914, 18sylbi 216 . . . . 5 ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
2019exlimiv 1926 . . . 4 (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
215, 20mpgbir 1794 . . 3 𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵})
22 df-ss 3964 . . 3 (({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}))
2321, 22mpbir 230 . 2 ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵}
24 df-he 43440 . 2 ({⟨𝐴, 𝐴⟩} hereditary {𝐵} ↔ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵})
2523, 24mpbir 230 1 {⟨𝐴, 𝐴⟩} hereditary {𝐵}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 394  w3a 1084  wal 1532   = wceq 1534  wex 1774  wcel 2099  wss 3947  {csn 4633  cop 4639  cima 5685   hereditary whe 43439
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697  ax-sep 5304  ax-nul 5311  ax-pr 5433
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3464  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-sn 4634  df-pr 4636  df-op 4640  df-br 5154  df-opab 5216  df-xp 5688  df-cnv 5690  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-he 43440
This theorem is referenced by: (None)
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