Theorem posn 5749

Description: Partial ordering of a singleton. (Contributed by NM, 27-Apr-2009.) (Revised by Mario Carneiro, 23-Apr-2015.)

Assertion

Ref	Expression
posn	⊢ (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Proof of Theorem posn

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		po0 5588	. . . . . 6 ⊢ 𝑅 Po ∅
2		snprc 4685	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3		poeq2 5575	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝑅 Po {𝐴} ↔ 𝑅 Po ∅))
4	2, 3	sylbi 220	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝑅 Po {𝐴} ↔ 𝑅 Po ∅))
5	1, 4	mpbiri 261	. . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝑅 Po {𝐴})
6	5	adantl 487	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → 𝑅 Po {𝐴})
7		brrelex1 5716	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐴) → 𝐴 ∈ V)
8	7	stoic1a 1805	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝐴)
9	6, 8	2thd 268	. . 3 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
10	9	ex 418	. 2 ⊢ (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴)))
11		df-po 5571	. . 3 ⊢ (𝑅 Po {𝐴} ↔ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
12		breq2 5115	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝐴))
13	12	anbi2d 642	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴)))
14		breq2 5115	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝑥𝑅𝑧 ↔ 𝑥𝑅𝐴))
15	13, 14	imbi12d 347	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴)))
16	15	anbi2d 642	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
17	16	ralsng 4643	. . . . . . 7 ⊢ (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
18	17	ralbidv 3190	. . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑦 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
19		simpl 488	. . . . . . . . . 10 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝑦)
20		breq2 5115	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐴))
21	19, 20	imbitrid 247	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))
22	21	biantrud 541	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (¬ 𝑥𝑅𝑥 ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
23	22	bicomd 226	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴)) ↔ ¬ 𝑥𝑅𝑥))
24	23	ralsng 4643	. . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴)) ↔ ¬ 𝑥𝑅𝑥))
25	18, 24	bitrd 282	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ¬ 𝑥𝑅𝑥))
26	25	ralbidv 3190	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥 ∈ {𝐴} ¬ 𝑥𝑅𝑥))
27		breq12 5116	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
28	27	anidms 577	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
29	28	notbid 321	. . . . 5 ⊢ (𝑥 = 𝐴 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝐴𝑅𝐴))
30	29	ralsng 4643	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} ¬ 𝑥𝑅𝑥 ↔ ¬ 𝐴𝑅𝐴))
31	26, 30	bitrd 282	. . 3 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ¬ 𝐴𝑅𝐴))
32	11, 31	bitrid 286	. 2 ⊢ (𝐴 ∈ V → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
33	10, 32	pm2.61d2 183	1 ⊢ (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∀wral 3081  Vcvv 3457  ∅c0 4286  {csn 4591   class class class wbr 5111   Po wpo 5569  Rel wrel 5668

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-po 5571  df-xp 5669  df-rel 5670

This theorem is used by:  sosn  5750