Theorem snhesn 44532

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 3459	. . . . . . 7 ⊢ 𝑥 ∈ V
2	1	elima3 6069	. . . . . 6 ⊢ (𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ↔ ∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}))
3		velsn 4605	. . . . . 6 ⊢ (𝑥 ∈ {𝐵} ↔ 𝑥 = 𝐵)
4	2, 3	imbi12i 353	. . . . 5 ⊢ ((𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
5	4	albii 1849	. . . 4 ⊢ (∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}) ↔ ∀𝑥(∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵))
6		velsn 4605	. . . . . . . 8 ⊢ (𝑦 ∈ {𝐵} ↔ 𝑦 = 𝐵)
7		opex 5445	. . . . . . . . . 10 ⊢ ⟨𝑦, 𝑥⟩ ∈ V
8	7	elsn 4604	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ ⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩)
9		vex 3459	. . . . . . . . . 10 ⊢ 𝑦 ∈ V
10	9, 1	opth 5458	. . . . . . . . 9 ⊢ (⟨𝑦, 𝑥⟩ = ⟨𝐴, 𝐴⟩ ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴))
11	8, 10	bitri 278	. . . . . . . 8 ⊢ (⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩} ↔ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴))
12	6, 11	anbi12i 639	. . . . . . 7 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴)))
13		3anass 1111	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) ↔ (𝑦 = 𝐵 ∧ (𝑦 = 𝐴 ∧ 𝑥 = 𝐴)))
14	12, 13	bitr4i 281	. . . . . 6 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) ↔ (𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴))
15		simp3 1156	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐴)
16		simp2 1155	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑦 = 𝐴)
17		simp1 1154	. . . . . . 7 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑦 = 𝐵)
18	15, 16, 17	3eqtr2d 2804	. . . . . 6 ⊢ ((𝑦 = 𝐵 ∧ 𝑦 = 𝐴 ∧ 𝑥 = 𝐴) → 𝑥 = 𝐵)
19	14, 18	sylbi 220	. . . . 5 ⊢ ((𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
20	19	exlimiv 1960	. . . 4 ⊢ (∃𝑦(𝑦 ∈ {𝐵} ∧ ⟨𝑦, 𝑥⟩ ∈ {⟨𝐴, 𝐴⟩}) → 𝑥 = 𝐵)
21	5, 20	mpgbir 1829	. . 3 ⊢ ∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵})
22		df-ss 3922	. . 3 ⊢ (({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵} ↔ ∀𝑥(𝑥 ∈ ({⟨𝐴, 𝐴⟩} “ {𝐵}) → 𝑥 ∈ {𝐵}))
23	21, 22	mpbir 234	. 2 ⊢ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵}
24		df-he 44519	. 2 ⊢ ({⟨𝐴, 𝐴⟩} hereditary {𝐵} ↔ ({⟨𝐴, 𝐴⟩} “ {𝐵}) ⊆ {𝐵})
25	23, 24	mpbir 234	1 ⊢ {⟨𝐴, 𝐴⟩} hereditary {𝐵}

Syntax hints:   → wi 4   ∧ wa 400   ∧ w3a 1103  ∀wal 1568   = wceq 1570  ∃wex 1809   ∈ wcel 2143   ⊆ wss 3905  {csn 4589  ⟨cop 4595   “ cima 5664   hereditary whe 44518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-xp 5667  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-he 44519

This theorem is referenced by: (None)