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Mirrors > Home > MPE Home > Th. List > ensn1OLD | Structured version Visualization version GIF version |
Description: Obsolete version of ensn1 9013 as of 23-Sep-2024. (Contributed by NM, 4-Nov-2002.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
ensn1OLD.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ensn1OLD | ⊢ {𝐴} ≈ 1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5430 | . . . 4 ⊢ {⟨𝐴, ∅⟩} ∈ V | |
2 | f1oeq1 6818 | . . . 4 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅})) | |
3 | ensn1OLD.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
4 | 0ex 5306 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 3, 4 | f1osn 6870 | . . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅} |
6 | 1, 2, 5 | ceqsexv2d 3528 | . . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅} |
7 | bren 8945 | . . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) | |
8 | 6, 7 | mpbir 230 | . 2 ⊢ {𝐴} ≈ {∅} |
9 | df1o2 8469 | . 2 ⊢ 1o = {∅} | |
10 | 8, 9 | breqtrri 5174 | 1 ⊢ {𝐴} ≈ 1o |
Colors of variables: wff setvar class |
Syntax hints: ∃wex 1781 ∈ wcel 2106 Vcvv 3474 ∅c0 4321 {csn 4627 ⟨cop 4633 class class class wbr 5147 –1-1-onto→wf1o 6539 1oc1o 8455 ≈ cen 8932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7721 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-mo 2534 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-suc 6367 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-1o 8462 df-en 8936 |
This theorem is referenced by: (None) |
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