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Mirrors > Home > MPE Home > Th. List > en1eqsn | Structured version Visualization version GIF version |
Description: A set with one element is a singleton. (Contributed by FL, 18-Aug-2008.) |
Ref | Expression |
---|---|
en1eqsn | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 8254 | . . . . . 6 ⊢ 1o ∈ ω | |
2 | ssid 3986 | . . . . . 6 ⊢ 1o ⊆ 1o | |
3 | ssnnfi 8725 | . . . . . 6 ⊢ ((1o ∈ ω ∧ 1o ⊆ 1o) → 1o ∈ Fin) | |
4 | 1, 2, 3 | mp2an 688 | . . . . 5 ⊢ 1o ∈ Fin |
5 | enfii 8723 | . . . . 5 ⊢ ((1o ∈ Fin ∧ 𝐵 ≈ 1o) → 𝐵 ∈ Fin) | |
6 | 4, 5 | mpan 686 | . . . 4 ⊢ (𝐵 ≈ 1o → 𝐵 ∈ Fin) |
7 | 6 | adantl 482 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 ∈ Fin) |
8 | snssi 4733 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
9 | 8 | adantr 481 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} ⊆ 𝐵) |
10 | ensn1g 8562 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ≈ 1o) | |
11 | ensym 8546 | . . . 4 ⊢ (𝐵 ≈ 1o → 1o ≈ 𝐵) | |
12 | entr 8549 | . . . 4 ⊢ (({𝐴} ≈ 1o ∧ 1o ≈ 𝐵) → {𝐴} ≈ 𝐵) | |
13 | 10, 11, 12 | syl2an 595 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} ≈ 𝐵) |
14 | fisseneq 8717 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ {𝐴} ⊆ 𝐵 ∧ {𝐴} ≈ 𝐵) → {𝐴} = 𝐵) | |
15 | 7, 9, 13, 14 | syl3anc 1363 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} = 𝐵) |
16 | 15 | eqcomd 2824 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∈ wcel 2105 ⊆ wss 3933 {csn 4557 class class class wbr 5057 ωcom 7569 1oc1o 8084 ≈ cen 8494 Fincfn 8497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-1o 8091 df-er 8278 df-en 8498 df-dom 8499 df-sdom 8500 df-fin 8501 |
This theorem is referenced by: en1eqsnbi 8737 1nsgtrivd 18264 gex1 18645 0cyg 18942 pgpfac1lem3a 19127 pgpfaclem3 19134 0ring 19971 en1top 21520 cnextfres1 22604 xrge0tsmseq 30621 sconnpi1 32383 rngoueqz 35099 isdmn3 35233 |
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