Theorem ipo0 44725

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 2014	. . . . 5 ⊢ 𝑥 = 𝑥
2		vex 3445	. . . . . 6 ⊢ 𝑥 ∈ V
3	2	ideq 5802	. . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥)
4	1, 3	mpbir 231	. . . 4 ⊢ 𝑥 I 𝑥
5		poirr 5545	. . . . 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 4360	. 2 ⊢ ( I Po 𝐴 → 𝐴 = ∅)
9		po0 5550	. . 3 ⊢ I Po ∅
10		poeq2 5537	. . 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 1542   ∈ wcel 2114  ∅c0 4286   class class class wbr 5099   I cid 5519   Po wpo 5531

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5242  ax-nul 5252  ax-pr 5378

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3062  df-rab 3401  df-v 3443  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4287  df-if 4481  df-sn 4582  df-pr 4584  df-op 4588  df-br 5100  df-opab 5162  df-id 5520  df-po 5533  df-xp 5631  df-rel 5632

This theorem is referenced by: (None)