Theorem posn 5711

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 5550	. . . . . 6 ⊢ 𝑅 Po ∅
2		snprc 4656	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅)
3		poeq2 5537	. . . . . . 7 ⊢ ({𝐴} = ∅ → (𝑅 Po {𝐴} ↔ 𝑅 Po ∅))
4	2, 3	sylbi 218	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (𝑅 Po {𝐴} ↔ 𝑅 Po ∅))
5	1, 4	mpbiri 259	. . . . 5 ⊢ (¬ 𝐴 ∈ V → 𝑅 Po {𝐴})
6	5	adantl 482	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → 𝑅 Po {𝐴})
7		brrelex1 5678	. . . . 5 ⊢ ((Rel 𝑅 ∧ 𝐴𝑅𝐴) → 𝐴 ∈ V)
8	7	stoic1a 1779	. . . 4 ⊢ ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝐴)
9	6, 8	2thd 266	. . 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 5533	. . 3 ⊢ (𝑅 Po {𝐴} ↔ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
12		breq2 5083	. . . . . . . . . . 11 ⊢ (𝑧 = 𝐴 → (𝑦𝑅𝑧 ↔ 𝑦𝑅𝐴))
13	12	anbi2d 636	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴)))
14		breq2 5083	. . . . . . . . . 10 ⊢ (𝑧 = 𝐴 → (𝑥𝑅𝑧 ↔ 𝑥𝑅𝐴))
15	13, 14	imbi12d 345	. . . . . . . . 9 ⊢ (𝑧 = 𝐴 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴)))
16	15	anbi2d 636	. . . . . . . 8 ⊢ (𝑧 = 𝐴 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
17	16	ralsng 4614	. . . . . . 7 ⊢ (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
18	17	ralbidv 3163	. . . . . 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 5083	. . . . . . . . . 10 ⊢ (𝑦 = 𝐴 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝐴))
21	19, 20	imbitrid 245	. . . . . . . . 9 ⊢ (𝑦 = 𝐴 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))
22	21	biantrud 536	. . . . . . . 8 ⊢ (𝑦 = 𝐴 → (¬ 𝑥𝑅𝑥 ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
23	22	bicomd 224	. . . . . . 7 ⊢ (𝑦 = 𝐴 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴)) ↔ ¬ 𝑥𝑅𝑥))
24	23	ralsng 4614	. . . . . 6 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝐴) → 𝑥𝑅𝐴)) ↔ ¬ 𝑥𝑅𝑥))
25	18, 24	bitrd 280	. . . . 5 ⊢ (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ¬ 𝑥𝑅𝑥))
26	25	ralbidv 3163	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥 ∈ {𝐴} ¬ 𝑥𝑅𝑥))
27		breq12 5084	. . . . . . 7 ⊢ ((𝑥 = 𝐴 ∧ 𝑥 = 𝐴) → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
28	27	anidms 571	. . . . . 6 ⊢ (𝑥 = 𝐴 → (𝑥𝑅𝑥 ↔ 𝐴𝑅𝐴))
29	28	notbid 319	. . . . 5 ⊢ (𝑥 = 𝐴 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝐴𝑅𝐴))
30	29	ralsng 4614	. . . 4 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} ¬ 𝑥𝑅𝑥 ↔ ¬ 𝐴𝑅𝐴))
31	26, 30	bitrd 280	. . 3 ⊢ (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ¬ 𝐴𝑅𝐴))
32	11, 31	bitrid 284	. 2 ⊢ (𝐴 ∈ V → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
33	10, 32	pm2.61d2 182	1 ⊢ (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   = wceq 1547   ∈ wcel 2119  ∀wral 3054  Vcvv 3432  ∅c0 4268  {csn 4562   class class class wbr 5079   Po wpo 5531  Rel wrel 5630

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-po 5533  df-xp 5631  df-rel 5632

This theorem is referenced by:  sosn  5712