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Mirrors > Home > MPE Home > Th. List > en0r | Structured version Visualization version GIF version |
Description: The empty set is equinumerous only to itself. (Contributed by BTernaryTau, 29-Nov-2024.) |
Ref | Expression |
---|---|
en0r | ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | encv 9001 | . . . . 5 ⊢ (∅ ≈ 𝐴 → (∅ ∈ V ∧ 𝐴 ∈ V)) | |
2 | breng 9002 | . . . . 5 ⊢ ((∅ ∈ V ∧ 𝐴 ∈ V) → (∅ ≈ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1-onto→𝐴)) | |
3 | 1, 2 | syl 17 | . . . 4 ⊢ (∅ ≈ 𝐴 → (∅ ≈ 𝐴 ↔ ∃𝑓 𝑓:∅–1-1-onto→𝐴)) |
4 | 3 | ibi 267 | . . 3 ⊢ (∅ ≈ 𝐴 → ∃𝑓 𝑓:∅–1-1-onto→𝐴) |
5 | f1o00 6891 | . . . . 5 ⊢ (𝑓:∅–1-1-onto→𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅)) | |
6 | 5 | simprbi 496 | . . . 4 ⊢ (𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅) |
7 | 6 | exlimiv 1930 | . . 3 ⊢ (∃𝑓 𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅) |
8 | 4, 7 | syl 17 | . 2 ⊢ (∅ ≈ 𝐴 → 𝐴 = ∅) |
9 | 0ex 5316 | . . . . 5 ⊢ ∅ ∈ V | |
10 | f1oeq1 6844 | . . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)) | |
11 | f1o0 6893 | . . . . 5 ⊢ ∅:∅–1-1-onto→∅ | |
12 | 9, 10, 11 | ceqsexv2d 3536 | . . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅ |
13 | breng 9002 | . . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)) | |
14 | 9, 9, 13 | mp2an 692 | . . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅) |
15 | 12, 14 | mpbir 231 | . . 3 ⊢ ∅ ≈ ∅ |
16 | breq2 5155 | . . 3 ⊢ (𝐴 = ∅ → (∅ ≈ 𝐴 ↔ ∅ ≈ ∅)) | |
17 | 15, 16 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → ∅ ≈ 𝐴) |
18 | 8, 17 | impbii 209 | 1 ⊢ (∅ ≈ 𝐴 ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 ∧ wa 395 = wceq 1539 ∃wex 1778 ∈ wcel 2108 Vcvv 3481 ∅c0 4342 class class class wbr 5151 –1-1-onto→wf1o 6568 ≈ cen 8990 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2177 ax-ext 2708 ax-sep 5305 ax-nul 5315 ax-pr 5441 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1779 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 3483 df-dif 3969 df-un 3971 df-ss 3983 df-nul 4343 df-if 4535 df-sn 4635 df-pr 4637 df-op 4641 df-br 5152 df-opab 5214 df-id 5587 df-xp 5699 df-rel 5700 df-cnv 5701 df-co 5702 df-dm 5703 df-rn 5704 df-fun 6571 df-fn 6572 df-f 6573 df-f1 6574 df-fo 6575 df-f1o 6576 df-en 8994 |
This theorem is referenced by: 0sdomg 9152 fiint 9373 rp-isfinite6 43524 ensucne0 43535 |
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