Theorem snhesn 38080

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 3203	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	elima3 5473	. . . . . 6 ⊢ (𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ↔ ∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}))
3		velsn 4193	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
4	2, 3	imbi12i 340	. . . . 5 ⊢ ((𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
5	4	albii 1747	. . . 4 ⊢ (∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
6		velsn 4193	. . . . . . . 8 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
7		opex 4932	. . . . . . . . . 10 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
8	7	elsn 4192	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩)
9		vex 3203	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
10	9, 1	opth 4945	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩ ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴))
11	8, 10	bitri 264	. . . . . . . 8 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴))
12	6, 11	anbi12i 733	. . . . . . 7 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴)))
13		3anass 1042	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴)))
14	12, 13	bitr4i 267	. . . . . 6 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴))
15		simp3 1063	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
16		simp2 1062	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑦 = 𝐴)
17		simp1 1061	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑦 = 𝐵)
18	15, 16, 17	3eqtr2d 2662	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵)
19	14, 18	sylbi 207	. . . . 5 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
20	19	exlimiv 1858	. . . 4 ⊢ (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
21	5, 20	mpgbir 1726	. . 3 ⊢ ∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵})
22		dfss2 3591	. . 3 ⊢ (({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}))
23	21, 22	mpbir 221	. 2 ⊢ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵}
24		df-he 38067	. 2 ⊢ ({⟨𝐴, 𝐴⟩} hereditary {𝐵} ↔ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵})
25	23, 24	mpbir 221	1 ⊢ {⟨𝐴, 𝐴⟩} hereditary {𝐵}

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990   ⊆ wss 3574  {csn 4177  ⟨cop 4183   “ cima 5117   hereditary whe 38066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-xp 5120  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-he 38067

This theorem is referenced by: (None)