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Mirrors > Home > MPE Home > Th. List > en0OLD | Structured version Visualization version GIF version |
Description: Obsolete version of en0 9042 as of 23-Sep-2024. (Contributed by NM, 27-May-1998.) Avoid ax-pow 5367. (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 8978 | . . 3 ⊢ (𝐴 ≈ ∅ ↔ ∃𝑓 𝑓:𝐴–1-1-onto→∅) | |
2 | f1ocnv 6854 | . . . . 5 ⊢ (𝑓:𝐴–1-1-onto→∅ → ◡𝑓:∅–1-1-onto→𝐴) | |
3 | f1o00 6877 | . . . . . 6 ⊢ (◡𝑓:∅–1-1-onto→𝐴 ↔ (◡𝑓 = ∅ ∧ 𝐴 = ∅)) | |
4 | 3 | simprbi 495 | . . . . 5 ⊢ (◡𝑓:∅–1-1-onto→𝐴 → 𝐴 = ∅) |
5 | 2, 4 | syl 17 | . . . 4 ⊢ (𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
6 | 5 | exlimiv 1925 | . . 3 ⊢ (∃𝑓 𝑓:𝐴–1-1-onto→∅ → 𝐴 = ∅) |
7 | 1, 6 | sylbi 216 | . 2 ⊢ (𝐴 ≈ ∅ → 𝐴 = ∅) |
8 | 0ex 5309 | . . . . 5 ⊢ ∅ ∈ V | |
9 | f1oeq1 6830 | . . . . 5 ⊢ (𝑓 = ∅ → (𝑓:∅–1-1-onto→∅ ↔ ∅:∅–1-1-onto→∅)) | |
10 | f1o0 6879 | . . . . 5 ⊢ ∅:∅–1-1-onto→∅ | |
11 | 8, 9, 10 | ceqsexv2d 3526 | . . . 4 ⊢ ∃𝑓 𝑓:∅–1-1-onto→∅ |
12 | bren 8978 | . . . 4 ⊢ (∅ ≈ ∅ ↔ ∃𝑓 𝑓:∅–1-1-onto→∅) | |
13 | 11, 12 | mpbir 230 | . . 3 ⊢ ∅ ≈ ∅ |
14 | breq1 5153 | . . 3 ⊢ (𝐴 = ∅ → (𝐴 ≈ ∅ ↔ ∅ ≈ ∅)) | |
15 | 13, 14 | mpbiri 257 | . 2 ⊢ (𝐴 = ∅ → 𝐴 ≈ ∅) |
16 | 7, 15 | impbii 208 | 1 ⊢ (𝐴 ≈ ∅ ↔ 𝐴 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1533 ∃wex 1773 ∅c0 4324 class class class wbr 5150 ◡ccnv 5679 –1-1-onto→wf1o 6550 ≈ cen 8965 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-12 2166 ax-ext 2698 ax-sep 5301 ax-nul 5308 ax-pr 5431 ax-un 7744 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-mo 2529 df-clab 2705 df-cleq 2719 df-clel 2805 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4325 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4911 df-br 5151 df-opab 5213 df-id 5578 df-xp 5686 df-rel 5687 df-cnv 5688 df-co 5689 df-dm 5690 df-rn 5691 df-fun 6553 df-fn 6554 df-f 6555 df-f1 6556 df-fo 6557 df-f1o 6558 df-en 8969 |
This theorem is referenced by: (None) |
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