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Mirrors > Home > MPE Home > Th. List > ensn1 | Structured version Visualization version GIF version |
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7721. (Revised by BTernaryTau, 23-Sep-2024.) |
Ref | Expression |
---|---|
ensn1.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ensn1 | ⊢ {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 | ensn1.1 | . . . . 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 | snex 5430 | . . . 4 ⊢ {𝐴} ∈ V | |
8 | snex 5430 | . . . 4 ⊢ {∅} ∈ V | |
9 | breng 8944 | . . . 4 ⊢ (({𝐴} ∈ V ∧ {∅} ∈ V) → ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})) | |
10 | 7, 8, 9 | mp2an 690 | . . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) |
11 | 6, 10 | mpbir 230 | . 2 ⊢ {𝐴} ≈ {∅} |
12 | df1o2 8469 | . 2 ⊢ 1o = {∅} | |
13 | 11, 12 | breqtrri 5174 | 1 ⊢ {𝐴} ≈ 1o |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃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 |
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-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: ensn1g 9015 en1 9017 en1OLD 9018 sdom1 9238 fodomfi 9321 pm54.43 9992 1nprm 16612 gex1 19453 sylow2a 19481 0frgp 19641 en1top 22478 en2top 22479 t1connperf 22931 ptcmplem2 23548 xrge0tsms2 24342 sconnpi1 34218 |
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