Theorem ipo0 45280

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 2045	. . . . 5 ⊢ 𝑥 = 𝑥
2		vex 3457	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	ideq 5836	. . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥)
4	1, 3	mpbir 234	. . . 4 ⊢ 𝑥 I 𝑥
5		poirr 5579	. . . . 5 ⊢ (( I Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥)
6	5	ex 418	. . . 4 ⊢ ( I Po 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥))
7	4, 6	mt2i 138	. . 3 ⊢ ( I Po 𝐴 → ¬ 𝑥 ∈ 𝐴)
8	7	eq0rdv 4368	. 2 ⊢ ( I Po 𝐴 → 𝐴 = ∅)
9		po0 5584	. . 3 ⊢ I Po ∅
10		poeq2 5571	. . 3 ⊢ (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅))
11	9, 10	mpbiri 261	. 2 ⊢ (𝐴 = ∅ → I Po 𝐴)
12	8, 11	impbii 212	1 ⊢ ( I Po 𝐴 ↔ 𝐴 = ∅)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   = wceq 1570   ∈ wcel 2145  ∅c0 4282   class class class wbr 5107   I cid 5553   Po wpo 5565

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 2147  ax-9 2155  ax-ext 2734  ax-sep 5255  ax-pr 5402

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 2741  df-cleq 2754  df-clel 2837  df-ral 3079  df-rex 3089  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-br 5108  df-opab 5172  df-id 5554  df-po 5567  df-xp 5665  df-rel 5666

This theorem is used by: (None)