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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 9013 | . . . . 5 ⊢ (∅ ≈ 𝐴 → (∅ ∈ V ∧ 𝐴 ∈ V)) | |
2 | breng 9014 | . . . . 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 6899 | . . . . 5 ⊢ (𝑓:∅–1-1-onto→𝐴 ↔ (𝑓 = ∅ ∧ 𝐴 = ∅)) | |
6 | 5 | simprbi 496 | . . . 4 ⊢ (𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅) |
7 | 6 | exlimiv 1929 | . . 3 ⊢ (∃𝑓 𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅) |
8 | 4, 7 | syl 17 | . 2 ⊢ (∅ ≈ 𝐴 → 𝐴 = ∅) |
9 | 0ex 5325 | . . . . 5 ⊢ ∅ ∈ V | |
10 | f1oeq1 6852 | . . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)) | |
11 | f1o0 6901 | . . . . 5 ⊢ ∅:∅–1-1-onto→∅ | |
12 | 9, 10, 11 | ceqsexv2d 3545 | . . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅ |
13 | breng 9014 | . . . . 5 ⊢ ((∅ ∈ V ∧ ∅ ∈ V) → (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅)) | |
14 | 9, 9, 13 | mp2an 691 | . . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅) |
15 | 12, 14 | mpbir 231 | . . 3 ⊢ ∅ ≈ ∅ |
16 | breq2 5170 | . . 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 1537 ∃wex 1777 ∈ wcel 2108 Vcvv 3488 ∅c0 4352 class class class wbr 5166 –1-1-onto→wf1o 6574 ≈ cen 9002 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1793 ax-4 1807 ax-5 1909 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-12 2178 ax-ext 2711 ax-sep 5317 ax-nul 5324 ax-pr 5447 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1089 df-tru 1540 df-fal 1550 df-ex 1778 df-sb 2065 df-mo 2543 df-clab 2718 df-cleq 2732 df-clel 2819 df-ral 3068 df-rex 3077 df-rab 3444 df-v 3490 df-dif 3979 df-un 3981 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-br 5167 df-opab 5229 df-id 5593 df-xp 5706 df-rel 5707 df-cnv 5708 df-co 5709 df-dm 5710 df-rn 5711 df-fun 6577 df-fn 6578 df-f 6579 df-f1 6580 df-fo 6581 df-f1o 6582 df-en 9006 |
This theorem is referenced by: 0sdomg 9172 fiint 9396 rp-isfinite6 43482 ensucne0 43493 |
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