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Theorem snhesn 38778
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 3353 . . . . . . 7 𝑥 ∈ V
21elima3 5657 . . . . . 6 (𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ↔ ∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}))
3 velsn 4352 . . . . . 6 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
42, 3imbi12i 341 . . . . 5 ((𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
54albii 1914 . . . 4 (∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
6 velsn 4352 . . . . . . . 8 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
7 opex 5090 . . . . . . . . . 10 𝑦, 𝑥⟩ ∈ V
87elsn 4351 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩)
9 vex 3353 . . . . . . . . . 10 𝑦 ∈ V
109, 1opth 5102 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩ ↔ (𝑦 = 𝐴𝑥 = 𝐴))
118, 10bitri 266 . . . . . . . 8 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ (𝑦 = 𝐴𝑥 = 𝐴))
126, 11anbi12i 620 . . . . . . 7 ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴𝑥 = 𝐴)))
13 3anass 1116 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴𝑥 = 𝐴)))
1412, 13bitr4i 269 . . . . . 6 ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴))
15 simp3 1168 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑥 = 𝐴)
16 simp2 1167 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑦 = 𝐴)
17 simp1 1166 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑦 = 𝐵)
1815, 16, 173eqtr2d 2805 . . . . . 6 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑥 = 𝐵)
1914, 18sylbi 208 . . . . 5 ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
2019exlimiv 2025 . . . 4 (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
215, 20mpgbir 1894 . . 3 𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵})
22 dfss2 3751 . . 3 (({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}))
2321, 22mpbir 222 . 2 ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵}
24 df-he 38765 . 2 ({⟨𝐴, 𝐴⟩} hereditary {𝐵} ↔ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵})
2523, 24mpbir 222 1 {⟨𝐴, 𝐴⟩} hereditary {𝐵}
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1107  wal 1650   = wceq 1652  wex 1874  wcel 2155  wss 3734  {csn 4336  cop 4342  cima 5282   hereditary whe 38764
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pr 5064
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-nul 4082  df-if 4246  df-sn 4337  df-pr 4339  df-op 4343  df-br 4812  df-opab 4874  df-xp 5285  df-cnv 5287  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-he 38765
This theorem is referenced by: (None)
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