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Mirrors > Home > MPE Home > Th. List > Mathboxes > ipo0 | Structured version Visualization version GIF version |
Description: If the identity relation partially orders any class, then that class is the null class. (Contributed by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
ipo0 | ⊢ ( I Po 𝐴 ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 2018 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | vex 3494 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | ideq 5716 | . . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥) |
4 | 1, 3 | mpbir 233 | . . . 4 ⊢ 𝑥 I 𝑥 |
5 | poirr 5478 | . . . . 5 ⊢ (( I Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥) | |
6 | 5 | ex 415 | . . . 4 ⊢ ( I Po 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥)) |
7 | 4, 6 | mt2i 139 | . . 3 ⊢ ( I Po 𝐴 → ¬ 𝑥 ∈ 𝐴) |
8 | 7 | eq0rdv 4350 | . 2 ⊢ ( I Po 𝐴 → 𝐴 = ∅) |
9 | po0 5483 | . . 3 ⊢ I Po ∅ | |
10 | poeq2 5471 | . . 3 ⊢ (𝐴 = ∅ → ( I Po 𝐴 ↔ I Po ∅)) | |
11 | 9, 10 | mpbiri 260 | . 2 ⊢ (𝐴 = ∅ → I Po 𝐴) |
12 | 8, 11 | impbii 211 | 1 ⊢ ( I Po 𝐴 ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1536 ∈ wcel 2113 ∅c0 4284 class class class wbr 5059 I cid 5452 Po wpo 5465 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pr 5323 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-nul 4285 df-if 4461 df-sn 4561 df-pr 4563 df-op 4567 df-br 5060 df-opab 5122 df-id 5453 df-po 5467 df-xp 5554 df-rel 5555 |
This theorem is referenced by: (None) |
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