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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 8259 | . . . . . 6 ⊢ 1o ∈ ω | |
2 | ssid 3989 | . . . . . 6 ⊢ 1o ⊆ 1o | |
3 | ssnnfi 8731 | . . . . . 6 ⊢ ((1o ∈ ω ∧ 1o ⊆ 1o) → 1o ∈ Fin) | |
4 | 1, 2, 3 | mp2an 690 | . . . . 5 ⊢ 1o ∈ Fin |
5 | enfii 8729 | . . . . 5 ⊢ ((1o ∈ Fin ∧ 𝐵 ≈ 1o) → 𝐵 ∈ Fin) | |
6 | 4, 5 | mpan 688 | . . . 4 ⊢ (𝐵 ≈ 1o → 𝐵 ∈ Fin) |
7 | 6 | adantl 484 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 ∈ Fin) |
8 | snssi 4735 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
9 | 8 | adantr 483 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} ⊆ 𝐵) |
10 | ensn1g 8568 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ≈ 1o) | |
11 | ensym 8552 | . . . 4 ⊢ (𝐵 ≈ 1o → 1o ≈ 𝐵) | |
12 | entr 8555 | . . . 4 ⊢ (({𝐴} ≈ 1o ∧ 1o ≈ 𝐵) → {𝐴} ≈ 𝐵) | |
13 | 10, 11, 12 | syl2an 597 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} ≈ 𝐵) |
14 | fisseneq 8723 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ {𝐴} ⊆ 𝐵 ∧ {𝐴} ≈ 𝐵) → {𝐴} = 𝐵) | |
15 | 7, 9, 13, 14 | syl3anc 1367 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} = 𝐵) |
16 | 15 | eqcomd 2827 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 = wceq 1533 ∈ wcel 2110 ⊆ wss 3936 {csn 4561 class class class wbr 5059 ωcom 7574 1oc1o 8089 ≈ cen 8500 Fincfn 8503 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3773 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-br 5060 df-opab 5122 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-om 7575 df-1o 8096 df-er 8283 df-en 8504 df-dom 8505 df-sdom 8506 df-fin 8507 |
This theorem is referenced by: en1eqsnbi 8743 1nsgtrivd 18320 gex1 18710 0cyg 19007 pgpfac1lem3a 19192 pgpfaclem3 19199 0ring 20037 en1top 21586 cnextfres1 22670 xrge0tsmseq 30689 sconnpi1 32481 rngoueqz 35212 isdmn3 35346 |
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