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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 5321, ax-un 7673. (Revised by BTernaryTau, 23-Sep-2024.) |
Ref | Expression |
---|---|
en0 | ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | encv 8894 | . . . . 5 ⊢ (𝐴 ≈ ∅ → (𝐴 ∈ V ∧ ∅ ∈ V)) | |
2 | breng 8895 | . . . . 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 6797 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴) | |
6 | f1o00 6820 | . . . . . 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 5265 | . . . . 5 ⊢ ∅ ∈ V | |
12 | f1oeq1 6773 | . . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)) | |
13 | f1o0 6822 | . . . . 5 ⊢ ∅:∅–1-1-onto→∅ | |
14 | 11, 12, 13 | ceqsexv2d 3496 | . . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅ |
15 | breng 8895 | . . . . 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 5109 | . . 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 3444 ∅c0 4283 class class class wbr 5106 ◡ccnv 5633 –1-1-onto→wf1o 6496 ≈ cen 8883 |
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 5257 ax-nul 5264 ax-pr 5385 |
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 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-en 8887 |
This theorem is referenced by: snfi 8991 enrefnn 8994 dom0 9049 dom0OLD 9050 0sdomgOLD 9052 sdom0 9055 findcard 9110 findcard2 9111 nneneq 9156 nneneqOLD 9168 snnen2oOLD 9174 enp1iOLD 9227 findcard2OLD 9231 fiint 9271 cantnff 9615 cantnf0 9616 cantnfp1lem2 9620 cantnflem1 9630 cantnf 9634 cnfcom2lem 9642 cardnueq0 9905 infmap2 10159 fin23lem26 10266 cardeq0 10493 hasheq0 14269 mreexexd 17533 pmtrfmvdn0 19249 pmtrsn 19306 rp-isfinite6 41878 ensucne0OLD 41890 |
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