![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > en0 | Structured version Visualization version GIF version |
Description: The empty set is equinumerous only to itself. Exercise 1 of [TakeutiZaring] p. 88. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5362, ax-un 7720. (Revised by BTernaryTau, 23-Sep-2024.) |
Ref | Expression |
---|---|
en0 | ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | encv 8943 | . . . . 5 ⊢ (𝐴 ≈ ∅ → (𝐴 ∈ V ∧ ∅ ∈ V)) | |
2 | breng 8944 | . . . . 5 ⊢ ((𝐴 ∈ V ∧ ∅ ∈ V) → (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (𝐴 ≈ ∅ → (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅)) |
4 | 3 | ibi 267 | . . 3 ⊢ (𝐴 ≈ ∅ → ∃𝑓 𝑓:𝐴–1-1-onto→∅) |
5 | f1ocnv 6842 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴) | |
6 | f1o00 6865 | . . . . . 6 ⊢ (◡𝑓:∅–1-1-onto→𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅)) | |
7 | 6 | simprbi 498 | . . . . 5 ⊢ (◡𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅) |
8 | 5, 7 | syl 17 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
9 | 8 | exlimiv 1934 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
10 | 4, 9 | syl 17 | . 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅) |
11 | 0ex 5306 | . . . . 5 ⊢ ∅ ∈ V | |
12 | f1oeq1 6818 | . . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)) | |
13 | f1o0 6867 | . . . . 5 ⊢ ∅:∅–1-1-onto→∅ | |
14 | 11, 12, 13 | ceqsexv2d 3528 | . . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅ |
15 | breng 8944 | . . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)) | |
16 | 11, 11, 15 | mp2an 691 | . . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅) |
17 | 14, 16 | mpbir 230 | . . 3 ⊢ ∅ ≈ ∅ |
18 | breq1 5150 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅)) | |
19 | 17, 18 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅) |
20 | 10, 19 | impbii 208 | 1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 Vcvv 3475 ∅c0 4321 class class class wbr 5147 ◡ccnv 5674 –1-1-onto→wf1o 6539 ≈ cen 8932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-12 2172 ax-ext 2704 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3063 df-rex 3072 df-rab 3434 df-v 3477 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-en 8936 |
This theorem is referenced by: snfi 9040 enrefnn 9043 dom0 9098 dom0OLD 9099 0sdomgOLD 9101 sdom0 9104 findcard 9159 findcard2 9160 nneneq 9205 nneneqOLD 9217 snnen2oOLD 9223 enp1iOLD 9276 findcard2OLD 9280 fiint 9320 cantnff 9665 cantnf0 9666 cantnfp1lem2 9670 cantnflem1 9680 cantnf 9684 cnfcom2lem 9692 cardnueq0 9955 infmap2 10209 fin23lem26 10316 cardeq0 10543 hasheq0 14319 mreexexd 17588 pmtrfmvdn0 19323 pmtrsn 19380 rp-isfinite6 42202 ensucne0OLD 42214 |
Copyright terms: Public domain | W3C validator |