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Theorem snhesn 38551
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 3331 . . . . . . 7 𝑥 ∈ V
21elima3 5619 . . . . . 6 (𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ↔ ∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}))
3 velsn 4325 . . . . . 6 (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
42, 3imbi12i 339 . . . . 5 ((𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
54albii 1884 . . . 4 (∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
6 velsn 4325 . . . . . . . 8 (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
7 opex 5069 . . . . . . . . . 10 𝑦, 𝑥⟩ ∈ V
87elsn 4324 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩)
9 vex 3331 . . . . . . . . . 10 𝑦 ∈ V
109, 1opth 5081 . . . . . . . . 9 (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩ ↔ (𝑦 = 𝐴𝑥 = 𝐴))
118, 10bitri 264 . . . . . . . 8 (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ (𝑦 = 𝐴𝑥 = 𝐴))
126, 11anbi12i 735 . . . . . . 7 ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴𝑥 = 𝐴)))
13 3anass 1081 . . . . . . 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 1130 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑥 = 𝐴)
16 simp2 1129 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑦 = 𝐴)
17 simp1 1128 . . . . . . 7 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑦 = 𝐵)
1815, 16, 173eqtr2d 2788 . . . . . 6 ((𝑦 = 𝐵𝑦 = 𝐴𝑥 = 𝐴) → 𝑥 = 𝐵)
1914, 18sylbi 207 . . . . 5 ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
2019exlimiv 1995 . . . 4 (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
215, 20mpgbir 1863 . . 3 𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵})
22 dfss2 3720 . . 3 (({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}))
2321, 22mpbir 221 . 2 ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵}
24 df-he 38538 . 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 383  w3a 1072  wal 1618   = wceq 1620  wex 1841  wcel 2127  wss 3703  {csn 4309  cop 4315  cima 5257   hereditary whe 38537
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1859  ax-4 1874  ax-5 1976  ax-6 2042  ax-7 2078  ax-9 2136  ax-10 2156  ax-11 2171  ax-12 2184  ax-13 2379  ax-ext 2728  ax-sep 4921  ax-nul 4929  ax-pr 5043
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1623  df-ex 1842  df-nf 1847  df-sb 2035  df-eu 2599  df-mo 2600  df-clab 2735  df-cleq 2741  df-clel 2744  df-nfc 2879  df-ral 3043  df-rex 3044  df-rab 3047  df-v 3330  df-dif 3706  df-un 3708  df-in 3710  df-ss 3717  df-nul 4047  df-if 4219  df-sn 4310  df-pr 4312  df-op 4316  df-br 4793  df-opab 4853  df-xp 5260  df-cnv 5262  df-dm 5264  df-rn 5265  df-res 5266  df-ima 5267  df-he 38538
This theorem is referenced by: (None)
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