Theorem ipo0 42438

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 2015	. . . . 5 ⊢ 𝑥 = 𝑥
2		vex 3446	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	ideq 5799	. . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥)
4	1, 3	mpbir 230	. . . 4 ⊢ 𝑥 I 𝑥
5		poirr 5549	. . . . 5 ⊢ (( I Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥)
6	5	ex 414	. . . 4 ⊢ ( I Po 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥))
7	4, 6	mt2i 137	. . 3 ⊢ ( I Po 𝐴 → ¬ 𝑥 ∈ 𝐴)
8	7	eq0rdv 4356	. 2 ⊢ ( I Po 𝐴 → 𝐴 = ∅)
9		po0 5554	. . 3 ⊢ I Po ∅
10		poeq2 5541	. . 3 ⊢ (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅))
11	9, 10	mpbiri 258	. 2 ⊢ (𝐴 = ∅ → I Po 𝐴)
12	8, 11	impbii 208	1 ⊢ ( I Po 𝐴 ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 205   = wceq 1541   ∈ wcel 2106  ∅c0 4274   class class class wbr 5097   I cid 5522   Po wpo 5535

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5248  ax-nul 5255  ax-pr 5377

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-sn 4579  df-pr 4581  df-op 4585  df-br 5098  df-opab 5160  df-id 5523  df-po 5537  df-xp 5631  df-rel 5632

This theorem is referenced by: (None)