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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 4130 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 1, 2 | f1osn 5501 | . . . 4 ⊢ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅} |
4 | 1, 2 | opex 4229 | . . . . . 6 ⊢ 〈𝐴, ∅〉 ∈ V |
5 | 4 | snex 4185 | . . . . 5 ⊢ {〈𝐴, ∅〉} ∈ V |
6 | f1oeq1 5449 | . . . . 5 ⊢ (𝑓 = {〈𝐴, ∅〉} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {〈𝐴, ∅〉}:{𝐴}–1-1-onto→{∅})) | |
7 | 5, 6 | spcev 2832 | . . . 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 6746 | . . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) | |
10 | 8, 9 | mpbir 146 | . 2 ⊢ {𝐴} ≈ {∅} |
11 | df1o2 6429 | . 2 ⊢ 1o = {∅} | |
12 | 10, 11 | breqtrri 4030 | 1 ⊢ {𝐴} ≈ 1o |
Colors of variables: wff set class |
Syntax hints: ∃wex 1492 ∈ wcel 2148 Vcvv 2737 ∅c0 3422 {csn 3592 〈cop 3595 class class class wbr 4003 –1-1-onto→wf1o 5215 1oc1o 6409 ≈ cen 6737 |
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 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 |
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 2739 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-br 4004 df-opab 4065 df-id 4293 df-suc 4371 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-1o 6416 df-en 6740 |
This theorem is referenced by: ensn1g 6796 en1 6798 pm54.43 7188 1nprm 12113 en1top 13513 |
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