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Mirrors > Home > ILE Home > Th. List > ensn1 | GIF version |
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) |
Ref | Expression |
---|---|
ensn1.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ensn1 | ⊢ {𝐴} ≈ 1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensn1.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
2 | 0ex 4132 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 1, 2 | f1osn 5503 | . . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅} |
4 | 1, 2 | opex 4231 | . . . . . 6 ⊢ ⟨𝐴, ∅⟩ ∈ V |
5 | 4 | snex 4187 | . . . . 5 ⊢ {⟨𝐴, ∅⟩} ∈ V |
6 | f1oeq1 5451 | . . . . 5 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅})) | |
7 | 5, 6 | spcev 2834 | . . . 4 ⊢ ({⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅} → ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) |
8 | 3, 7 | ax-mp 5 | . . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅} |
9 | bren 6749 | . . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) | |
10 | 8, 9 | mpbir 146 | . 2 ⊢ {𝐴} ≈ {∅} |
11 | df1o2 6432 | . 2 ⊢ 1o = {∅} | |
12 | 10, 11 | breqtrri 4032 | 1 ⊢ {𝐴} ≈ 1o |
Colors of variables: wff set class |
Syntax hints: ∃wex 1492 ∈ wcel 2148 Vcvv 2739 ∅c0 3424 {csn 3594 ⟨cop 3597 class class class wbr 4005 –1-1-onto→wf1o 5217 1oc1o 6412 ≈ cen 6740 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-nul 4131 ax-pow 4176 ax-pr 4211 ax-un 4435 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-v 2741 df-dif 3133 df-un 3135 df-in 3137 df-ss 3144 df-nul 3425 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-id 4295 df-suc 4373 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-fun 5220 df-fn 5221 df-f 5222 df-f1 5223 df-fo 5224 df-f1o 5225 df-1o 6419 df-en 6743 |
This theorem is referenced by: ensn1g 6799 en1 6801 pm54.43 7191 1nprm 12116 en1top 13662 |
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