![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
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 6761 | . . 3 ⊢ (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅) | |
2 | f1ocnv 5486 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴) | |
3 | f1o00 5508 | . . . . . 6 ⊢ (◡𝑓:∅–1-1-onto→𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅)) | |
4 | 3 | simprbi 275 | . . . . 5 ⊢ (◡𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅) |
5 | 2, 4 | syl 14 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
6 | 5 | exlimiv 1608 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
7 | 1, 6 | sylbi 121 | . 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅) |
8 | 0ex 4142 | . . . 4 ⊢ ∅ ∈ V | |
9 | 8 | enref 6779 | . . 3 ⊢ ∅ ≈ ∅ |
10 | breq1 4018 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅)) | |
11 | 9, 10 | mpbiri 168 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅) |
12 | 7, 11 | impbii 126 | 1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff set class |
Syntax hints: ↔ wb 105 = wceq 1363 ∃wex 1502 ∅c0 3434 class class class wbr 4015 ◡ccnv 4637 –1-1-onto→wf1o 5227 ≈ cen 6752 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 615 ax-in2 616 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-nul 4141 ax-pow 4186 ax-pr 4221 ax-un 4445 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-fal 1369 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-dif 3143 df-un 3145 df-in 3147 df-ss 3154 df-nul 3435 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-res 4650 df-ima 4651 df-fun 5230 df-fn 5231 df-f 5232 df-f1 5233 df-fo 5234 df-f1o 5235 df-en 6755 |
This theorem is referenced by: nneneq 6871 php5 6872 snnen2oprc 6874 php5dom 6877 ssfilem 6889 dif1enen 6894 fin0 6899 fin0or 6900 diffitest 6901 findcard 6902 findcard2 6903 findcard2s 6904 diffisn 6907 fiintim 6942 fisseneq 6945 fihasheq0 10787 zfz1iso 10835 |
Copyright terms: Public domain | W3C validator |