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Mirrors > Home > MPE Home > Th. List > en0OLD | Structured version Visualization version GIF version |
Description: Obsolete version of en0 9080 as of 23-Sep-2024. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5383. (Revised by BTernaryTau, 31-Jul-2024.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
en0OLD | ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bren 9015 | . . 3 ⊢ (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅) | |
2 | f1ocnv 6876 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴) | |
3 | f1o00 6899 | . . . . . 6 ⊢ (◡𝑓:∅–1-1-onto→𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅)) | |
4 | 3 | simprbi 496 | . . . . 5 ⊢ (◡𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
6 | 5 | exlimiv 1929 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
7 | 1, 6 | sylbi 217 | . 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅) |
8 | 0ex 5325 | . . . . 5 ⊢ ∅ ∈ V | |
9 | f1oeq1 6852 | . . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)) | |
10 | f1o0 6901 | . . . . 5 ⊢ ∅:∅–1-1-onto→∅ | |
11 | 8, 9, 10 | ceqsexv2d 3545 | . . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅ |
12 | bren 9015 | . . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅) | |
13 | 11, 12 | mpbir 231 | . . 3 ⊢ ∅ ≈ ∅ |
14 | breq1 5169 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅)) | |
15 | 13, 14 | mpbiri 258 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅) |
16 | 7, 15 | impbii 209 | 1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 206 = wceq 1537 ∃wex 1777 ∅c0 4352 class class class wbr 5166 ◡ccnv 5699 –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 ax-un 7772 |
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-in 3983 df-ss 3993 df-nul 4353 df-if 4549 df-sn 4649 df-pr 4651 df-op 4655 df-uni 4932 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: (None) |
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