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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 5365, ax-un 7755. (Revised by BTernaryTau, 23-Sep-2024.) |
| Ref | Expression |
|---|---|
| en0 | ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | encv 8993 | . . . . 5 ⊢ (𝐴 ≈ ∅ → (𝐴 ∈ V ∧ ∅ ∈ V)) | |
| 2 | breng 8994 | . . . . 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 6860 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴) | |
| 6 | f1o00 6883 | . . . . . 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 1930 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
| 10 | 4, 9 | syl 17 | . 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅) |
| 11 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 12 | f1oeq1 6836 | . . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)) | |
| 13 | f1o0 6885 | . . . . 5 ⊢ ∅:∅–1-1-onto→∅ | |
| 14 | 11, 12, 13 | ceqsexv2d 3533 | . . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅ |
| 15 | breng 8994 | . . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)) | |
| 16 | 11, 11, 15 | mp2an 692 | . . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅) |
| 17 | 14, 16 | mpbir 231 | . . 3 ⊢ ∅ ≈ ∅ |
| 18 | breq1 5146 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅)) | |
| 19 | 17, 18 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅) |
| 20 | 10, 19 | impbii 209 | 1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 class class class wbr 5143 ◡ccnv 5684 –1-1-onto→wf1o 6560 ≈ cen 8982 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-en 8986 |
| This theorem is referenced by: 0fi 9082 snfiOLD 9084 enrefnn 9087 dom0 9142 dom0OLD 9143 0sdomgOLD 9145 sdom0 9148 findcard 9203 findcard2 9204 nneneq 9246 nneneqOLD 9258 snnen2oOLD 9264 enp1iOLD 9314 fiintOLD 9367 cantnff 9714 cantnf0 9715 cantnfp1lem2 9719 cantnflem1 9729 cantnf 9733 cnfcom2lem 9741 cardnueq0 10004 infmap2 10257 fin23lem26 10365 cardeq0 10592 hasheq0 14402 mreexexd 17691 pmtrfmvdn0 19480 pmtrsn 19537 rp-isfinite6 43531 ensucne0OLD 43543 |
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