Theorem ipo0 44413

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 2011	. . . . 5 ⊢ 𝑥 = 𝑥
2		vex 3492	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	ideq 5872	. . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥)
4	1, 3	mpbir 231	. . . 4 ⊢ 𝑥 I 𝑥
5		poirr 5619	. . . . 5 ⊢ (( I Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥)
6	5	ex 412	. . . 4 ⊢ ( I Po 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥))
7	4, 6	mt2i 137	. . 3 ⊢ ( I Po 𝐴 → ¬ 𝑥 ∈ 𝐴)
8	7	eq0rdv 4430	. 2 ⊢ ( I Po 𝐴 → 𝐴 = ∅)
9		po0 5624	. . 3 ⊢ I Po ∅
10		poeq2 5611	. . 3 ⊢ (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅))
11	9, 10	mpbiri 258	. 2 ⊢ (𝐴 = ∅ → I Po 𝐴)
12	8, 11	impbii 209	1 ⊢ ( I Po 𝐴 ↔ 𝐴 = ∅)

Syntax hints:  ¬ wn 3   ↔ wb 206   = wceq 1537   ∈ wcel 2108  ∅c0 4352   class class class wbr 5166   I cid 5592   Po wpo 5605

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-id 5593  df-po 5607  df-xp 5701  df-rel 5702

This theorem is referenced by: (None)