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Mirrors > Home > ILE Home > Th. List > en0 | GIF version |
Description: The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) |
Ref | Expression |
---|---|
en0 | ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren 6634 | . . 3 ⊢ (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅) | |
2 | f1ocnv 5373 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴) | |
3 | f1o00 5395 | . . . . . 6 ⊢ (◡𝑓:∅–1-1-onto→𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅)) | |
4 | 3 | simprbi 273 | . . . . 5 ⊢ (◡𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅) |
5 | 2, 4 | syl 14 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
6 | 5 | exlimiv 1577 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
7 | 1, 6 | sylbi 120 | . 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅) |
8 | 0ex 4050 | . . . 4 ⊢ ∅ ∈ V | |
9 | 8 | enref 6652 | . . 3 ⊢ ∅ ≈ ∅ |
10 | breq1 3927 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅)) | |
11 | 9, 10 | mpbiri 167 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅) |
12 | 7, 11 | impbii 125 | 1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 104 = wceq 1331 ∃wex 1468 ∅c0 3358 class class class wbr 3924 ◡ccnv 4533 –1-1-onto→wf1o 5117 ≈ cen 6625 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ral 2419 df-rex 2420 df-v 2683 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-en 6628 |
This theorem is referenced by: nneneq 6744 php5 6745 snnen2oprc 6747 php5dom 6750 ssfilem 6762 dif1enen 6767 fin0 6772 fin0or 6773 diffitest 6774 findcard 6775 findcard2 6776 findcard2s 6777 diffisn 6780 fiintim 6810 fisseneq 6813 fihasheq0 10533 zfz1iso 10577 |
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