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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 2019 | . . . . 5 ⊢ 𝑥 = 𝑥 | |
2 | vex 3499 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | ideq 5725 | . . . . 5 ⊢ (𝑥 I 𝑥 ↔ 𝑥 = 𝑥) |
4 | 1, 3 | mpbir 233 | . . . 4 ⊢ 𝑥 I 𝑥 |
5 | frirr 5534 | . . . . 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 4359 | . 2 ⊢ ( I Fr 𝐴 → 𝐴 = ∅) |
9 | fr0 5536 | . . 3 ⊢ I Fr ∅ | |
10 | freq2 5528 | . . 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 1537 ∈ wcel 2114 ∅c0 4293 class class class wbr 5068 I cid 5461 Fr wfr 5513 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pr 5332 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-sn 4570 df-pr 4572 df-op 4576 df-br 5069 df-opab 5131 df-id 5462 df-fr 5516 df-xp 5563 df-rel 5564 |
This theorem is referenced by: (None) |
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