Theorem posn 5672

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 5520	. . . . . 6 ⊢ 𝑅 Po ∅
2		snprc 4653	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3		poeq2 5507	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝑅 Po {𝐴} ↔ 𝑅 Po ∅))
4	2, 3	sylbi 216	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝑅 Po {𝐴} ↔ 𝑅 Po ∅))
5	1, 4	mpbiri 257	. . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝑅 Po {𝐴})
6	5	adantl 482	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → 𝑅 Po {𝐴})
7		brrelex1 5640	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐴) → 𝐴 ∈ V)
8	7	stoic1a 1775	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝐴)
9	6, 8	2thd 264	. . 3 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
10	9	ex 413	. 2 ⊢ (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴)))
11		df-po 5503	. . 3 ⊢ (𝑅 Po {𝐴} ↔ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
12		breq2 5078	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝐴))
13	12	anbi2d 629	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴)))
14		breq2 5078	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝑥𝑅𝑧 ↔ 𝑥𝑅𝐴))
15	13, 14	imbi12d 345	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴)))
16	15	anbi2d 629	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
17	16	ralsng 4609	. . . . . . 7 ⊢ (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
18	17	ralbidv 3112	. . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑦 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
19		simpl 483	. . . . . . . . . 10 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝑦)
20		breq2 5078	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐴))
21	19, 20	syl5ib 243	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))
22	21	biantrud 532	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (¬ 𝑥𝑅𝑥 ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
23	22	bicomd 222	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴)) ↔ ¬ 𝑥𝑅𝑥))
24	23	ralsng 4609	. . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴)) ↔ ¬ 𝑥𝑅𝑥))
25	18, 24	bitrd 278	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ¬ 𝑥𝑅𝑥))
26	25	ralbidv 3112	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥 ∈ {𝐴} ¬ 𝑥𝑅𝑥))
27		breq12 5079	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
28	27	anidms 567	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
29	28	notbid 318	. . . . 5 ⊢ (𝑥 = 𝐴 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝐴𝑅𝐴))
30	29	ralsng 4609	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} ¬ 𝑥𝑅𝑥 ↔ ¬ 𝐴𝑅𝐴))
31	26, 30	bitrd 278	. . 3 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ¬ 𝐴𝑅𝐴))
32	11, 31	bitrid 282	. 2 ⊢ (𝐴 ∈ V → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
33	10, 32	pm2.61d2 181	1 ⊢ (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106  ∀wral 3064  Vcvv 3432  ∅c0 4256  {csn 4561   class class class wbr 5074   Po wpo 5501  Rel wrel 5594

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-po 5503  df-xp 5595  df-rel 5596

This theorem is referenced by:  sosn  5673