Theorem ipo0 39433

Description: If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ipo0	⊢ ( I Po 𝐴 ↔ 𝐴 = ∅)

Proof of Theorem ipo0

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		equid 2111	. . . . 5 ⊢ 𝑥 = 𝑥
2		vex 3388	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	ideq 5478	. . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥)
4	1, 3	mpbir 223	. . . 4 ⊢ 𝑥 I 𝑥
5		poirr 5244	. . . . 5 ⊢ (( I Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥)
6	5	ex 402	. . . 4 ⊢ ( I Po 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥))
7	4, 6	mt2i 135	. . 3 ⊢ ( I Po 𝐴 → ¬ 𝑥 ∈ 𝐴)
8	7	eq0rdv 4175	. 2 ⊢ ( I Po 𝐴 → 𝐴 = ∅)
9		po0 5248	. . 3 ⊢ I Po ∅
10		poeq2 5237	. . 3 ⊢ (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅))
11	9, 10	mpbiri 250	. 2 ⊢ (𝐴 = ∅ → I Po 𝐴)
12	8, 11	impbii 201	1 ⊢ ( I Po 𝐴 ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 198   = wceq 1653   ∈ wcel 2157  ∅c0 4115   class class class wbr 4843   I cid 5219   Po wpo 5231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-id 5220  df-po 5233  df-xp 5318  df-rel 5319

This theorem is referenced by: (None)