Theorem snhesn 41394

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.

Step	Hyp	Ref	Expression
1		vex 3436	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	elima3 5976	. . . . . 6 ⊢ (𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ↔ ∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}))
3		velsn 4577	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
4	2, 3	imbi12i 351	. . . . 5 ⊢ ((𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
5	4	albii 1822	. . . 4 ⊢ (∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
6		velsn 4577	. . . . . . . 8 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
7		opex 5379	. . . . . . . . . 10 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
8	7	elsn 4576	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩)
9		vex 3436	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
10	9, 1	opth 5391	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩ ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴))
11	8, 10	bitri 274	. . . . . . . 8 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴))
12	6, 11	anbi12i 627	. . . . . . 7 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴)))
13		3anass 1094	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴)))
14	12, 13	bitr4i 277	. . . . . 6 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴))
15		simp3 1137	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
16		simp2 1136	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑦 = 𝐴)
17		simp1 1135	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑦 = 𝐵)
18	15, 16, 17	3eqtr2d 2784	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵)
19	14, 18	sylbi 216	. . . . 5 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
20	19	exlimiv 1933	. . . 4 ⊢ (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
21	5, 20	mpgbir 1802	. . 3 ⊢ ∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵})
22		dfss2 3907	. . 3 ⊢ (({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}))
23	21, 22	mpbir 230	. 2 ⊢ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵}
24		df-he 41381	. 2 ⊢ ({⟨𝐴, 𝐴⟩} hereditary {𝐵} ↔ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵})
25	23, 24	mpbir 230	1 ⊢ {⟨𝐴, 𝐴⟩} hereditary {𝐵}

Syntax hints:   → wi 4   ∧ wa 396   ∧ w3a 1086  ∀wal 1537   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ⊆ wss 3887  {csn 4561  ⟨cop 4567   “ cima 5592   hereditary whe 41380

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-xp 5595  df-cnv 5597  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-he 41381

This theorem is referenced by: (None)