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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 7755. (Revised by BTernaryTau, 23-Sep-2024.) |
| Ref | Expression |
|---|---|
| ensn1.1 | ⊢ 𝐴 ∈ V |
| Ref | Expression |
|---|---|
| ensn1 | ⊢ {𝐴} ≈ 1o |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | snex 5436 | . . . 4 ⊢ {〈𝐴, ∅〉} ∈ V | |
| 2 | f1oeq1 6836 | . . . 4 ⊢ (𝑓 = {〈𝐴, ∅〉} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅})) | |
| 3 | ensn1.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
| 4 | 0ex 5307 | . . . . 5 ⊢ ∅ ∈ V | |
| 5 | 3, 4 | f1osn 6888 | . . . 4 ⊢ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅} |
| 6 | 1, 2, 5 | ceqsexv2d 3533 | . . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅} |
| 7 | snex 5436 | . . . 4 ⊢ {𝐴} ∈ V | |
| 8 | snex 5436 | . . . 4 ⊢ {∅} ∈ V | |
| 9 | breng 8994 | . . . 4 ⊢ (({𝐴} ∈ V ∧ {∅} ∈ V) → ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})) | |
| 10 | 7, 8, 9 | mp2an 692 | . . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) |
| 11 | 6, 10 | mpbir 231 | . 2 ⊢ {𝐴} ≈ {∅} |
| 12 | df1o2 8513 | . 2 ⊢ 1o = {∅} | |
| 13 | 11, 12 | breqtrri 5170 | 1 ⊢ {𝐴} ≈ 1o |
| Colors of variables: wff setvar class |
| Syntax hints: ↔ wb 206 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 〈cop 4632 class class class wbr 5143 –1-1-onto→wf1o 6560 1oc1o 8499 ≈ cen 8982 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-sb 2065 df-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-suc 6390 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-1o 8506 df-en 8986 |
| This theorem is referenced by: ensn1g 9062 en1 9064 sdom1 9278 fodomfiOLD 9370 pm54.43 10041 1nprm 16716 gex1 19609 sylow2a 19637 0frgp 19797 en1top 22991 en2top 22992 t1connperf 23444 ptcmplem2 24061 xrge0tsms2 24857 sconnpi1 35244 setcsnterm 49133 |
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