Mathbox for Andrew Salmon |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > Mathboxes > ifr0 | Structured version Visualization version GIF version |
Description: A class that is founded by the identity relation is null. (Contributed by Andrew Salmon, 25-Jul-2011.) |
Ref | Expression |
---|---|
ifr0 | ⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | equid 2015 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | vex 3497 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | ideq 5722 | . . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥) |
4 | 1, 3 | mpbir 233 | . . . 4 ⊢ 𝑥 I 𝑥 |
5 | frirr 5531 | . . . . 5 ⊢ (( I Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥) | |
6 | 5 | ex 415 | . . . 4 ⊢ ( I Fr 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥)) |
7 | 4, 6 | mt2i 139 | . . 3 ⊢ ( I Fr 𝐴 → ¬ 𝑥 ∈ 𝐴) |
8 | 7 | eq0rdv 4356 | . 2 ⊢ ( I Fr 𝐴 → 𝐴 = ∅) |
9 | fr0 5533 | . . 3 ⊢ I Fr ∅ | |
10 | freq2 5525 | . . 3 ⊢ (𝐴 = ∅ → ( I Fr 𝐴 ↔ I Fr ∅)) | |
11 | 9, 10 | mpbiri 260 | . 2 ⊢ (𝐴 = ∅ → I Fr 𝐴) |
12 | 8, 11 | impbii 211 | 1 ⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 208 = wceq 1533 ∈ wcel 2110 ∅c0 4290 class class class wbr 5065 I cid 5458 Fr wfr 5510 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-sep 5202 ax-nul 5209 ax-pr 5329 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-br 5066 df-opab 5128 df-id 5459 df-fr 5513 df-xp 5560 df-rel 5561 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |