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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 5359, ax-un 7734. (Revised by BTernaryTau, 23-Sep-2024.) |
Ref | Expression |
---|---|
en0 | ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | encv 8963 | . . . . 5 ⊢ (𝐴 ≈ ∅ → (𝐴 ∈ V ∧ ∅ ∈ V)) | |
2 | breng 8964 | . . . . 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 6845 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴) | |
6 | f1o00 6868 | . . . . . 6 ⊢ (◡𝑓:∅–1-1-onto→𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅)) | |
7 | 6 | simprbi 496 | . . . . 5 ⊢ (◡𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅) |
8 | 5, 7 | syl 17 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
9 | 8 | exlimiv 1926 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
10 | 4, 9 | syl 17 | . 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅) |
11 | 0ex 5301 | . . . . 5 ⊢ ∅ ∈ V | |
12 | f1oeq1 6821 | . . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)) | |
13 | f1o0 6870 | . . . . 5 ⊢ ∅:∅–1-1-onto→∅ | |
14 | 11, 12, 13 | ceqsexv2d 3524 | . . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅ |
15 | breng 8964 | . . . . 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 5145 | . . 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 395 = wceq 1534 ∃wex 1774 ∈ wcel 2099 Vcvv 3469 ∅c0 4318 class class class wbr 5142 ◡ccnv 5671 –1-1-onto→wf1o 6541 ≈ cen 8952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-sb 2061 df-mo 2529 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-br 5143 df-opab 5205 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-en 8956 |
This theorem is referenced by: snfi 9060 enrefnn 9063 dom0 9118 dom0OLD 9119 0sdomgOLD 9121 sdom0 9124 findcard 9179 findcard2 9180 nneneq 9225 nneneqOLD 9237 snnen2oOLD 9243 enp1iOLD 9296 findcard2OLD 9300 fiint 9340 cantnff 9689 cantnf0 9690 cantnfp1lem2 9694 cantnflem1 9704 cantnf 9708 cnfcom2lem 9716 cardnueq0 9979 infmap2 10233 fin23lem26 10340 cardeq0 10567 hasheq0 14346 mreexexd 17619 pmtrfmvdn0 19408 pmtrsn 19465 rp-isfinite6 42871 ensucne0OLD 42883 |
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