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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 1926 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | vex 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | ideq 5196 | . . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥) |
4 | 1, 3 | mpbir 220 | . . . 4 ⊢ 𝑥 I 𝑥 |
5 | frirr 5015 | . . . . 5 ⊢ (( I Fr 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥 I 𝑥) | |
6 | 5 | ex 449 | . . . 4 ⊢ ( I Fr 𝐴 → (𝑥 ∈ 𝐴 → ¬ 𝑥 I 𝑥)) |
7 | 4, 6 | mt2i 131 | . . 3 ⊢ ( I Fr 𝐴 → ¬ 𝑥 ∈ 𝐴) |
8 | 7 | eq0rdv 3931 | . 2 ⊢ ( I Fr 𝐴 → 𝐴 = ∅) |
9 | fr0 5017 | . . 3 ⊢ I Fr ∅ | |
10 | freq2 5009 | . . 3 ⊢ (𝐴 = ∅ → ( I Fr 𝐴 ↔ I Fr ∅)) | |
11 | 9, 10 | mpbiri 247 | . 2 ⊢ (𝐴 = ∅ → I Fr 𝐴) |
12 | 8, 11 | impbii 198 | 1 ⊢ ( I Fr 𝐴 ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 195 = wceq 1475 ∈ wcel 1977 ∅c0 3874 class class class wbr 4583 I cid 4948 Fr wfr 4994 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-id 4953 df-fr 4997 df-xp 5044 df-rel 5045 |
This theorem is referenced by: (None) |
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