Theorem snhesn 44213

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 3433	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	elima3 6032	. . . . . 6 ⊢ (𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ↔ ∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}))
3		velsn 4583	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
4	2, 3	imbi12i 350	. . . . 5 ⊢ ((𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
5	4	albii 1821	. . . 4 ⊢ (∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
6		velsn 4583	. . . . . . . 8 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
7		opex 5416	. . . . . . . . . 10 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
8	7	elsn 4582	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩)
9		vex 3433	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
10	9, 1	opth 5429	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩ ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴))
11	8, 10	bitri 275	. . . . . . . 8 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴))
12	6, 11	anbi12i 629	. . . . . . 7 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴)))
13		3anass 1095	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴)))
14	12, 13	bitr4i 278	. . . . . 6 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴))
15		simp3 1139	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
16		simp2 1138	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑦 = 𝐴)
17		simp1 1137	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑦 = 𝐵)
18	15, 16, 17	3eqtr2d 2777	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵)
19	14, 18	sylbi 217	. . . . 5 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
20	19	exlimiv 1932	. . . 4 ⊢ (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
21	5, 20	mpgbir 1801	. . 3 ⊢ ∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵})
22		df-ss 3906	. . 3 ⊢ (({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}))
23	21, 22	mpbir 231	. 2 ⊢ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵}
24		df-he 44200	. 2 ⊢ ({⟨𝐴, 𝐴⟩} hereditary {𝐵} ↔ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵})
25	23, 24	mpbir 231	1 ⊢ {⟨𝐴, 𝐴⟩} hereditary {𝐵}

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1087  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114   ⊆ wss 3889  {csn 4567  ⟨cop 4573   “ cima 5634   hereditary whe 44199

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-he 44200

This theorem is referenced by: (None)