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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 7673. (Revised by BTernaryTau, 23-Sep-2024.) |
Ref | Expression |
---|---|
ensn1.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
ensn1 | ⊢ {𝐴} ≈ 1o |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | snex 5389 | . . . 4 ⊢ {⟨𝐴, ∅⟩} ∈ V | |
2 | f1oeq1 6773 | . . . 4 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅})) | |
3 | ensn1.1 | . . . . 5 ⊢ 𝐴 ∈ V | |
4 | 0ex 5265 | . . . . 5 ⊢ ∅ ∈ V | |
5 | 3, 4 | f1osn 6825 | . . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅} |
6 | 1, 2, 5 | ceqsexv2d 3496 | . . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅} |
7 | snex 5389 | . . . 4 ⊢ {𝐴} ∈ V | |
8 | snex 5389 | . . . 4 ⊢ {∅} ∈ V | |
9 | breng 8895 | . . . 4 ⊢ (({𝐴} ∈ V ∧ {∅} ∈ V) → ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})) | |
10 | 7, 8, 9 | mp2an 691 | . . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}) |
11 | 6, 10 | mpbir 230 | . 2 ⊢ {𝐴} ≈ {∅} |
12 | df1o2 8420 | . 2 ⊢ 1o = {∅} | |
13 | 11, 12 | breqtrri 5133 | 1 ⊢ {𝐴} ≈ 1o |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∃wex 1782 ∈ wcel 2107 Vcvv 3444 ∅c0 4283 {csn 4587 ⟨cop 4593 class class class wbr 5106 –1-1-onto→wf1o 6496 1oc1o 8406 ≈ cen 8883 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-mo 2535 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-suc 6324 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-1o 8413 df-en 8887 |
This theorem is referenced by: ensn1g 8966 en1 8968 en1OLD 8969 sdom1 9189 fodomfi 9272 pm54.43 9942 1nprm 16560 gex1 19378 sylow2a 19406 0frgp 19566 en1top 22350 en2top 22351 t1connperf 22803 ptcmplem2 23420 xrge0tsms2 24214 sconnpi1 33890 |
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