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Mirrors > Home > ILE Home > Th. List > en1eqsn | 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 6520 | . . . . . 6 ⊢ 1o ∈ ω | |
2 | nnfi 6871 | . . . . . 6 ⊢ (1o ∈ ω → 1o ∈ Fin) | |
3 | 1, 2 | ax-mp 5 | . . . . 5 ⊢ 1o ∈ Fin |
4 | enfii 6873 | . . . . 5 ⊢ ((1o ∈ Fin ∧ 𝐵 ≈ 1o) → 𝐵 ∈ Fin) | |
5 | 3, 4 | mpan 424 | . . . 4 ⊢ (𝐵 ≈ 1o → 𝐵 ∈ Fin) |
6 | 5 | adantl 277 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 ∈ Fin) |
7 | snssi 3736 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ⊆ 𝐵) | |
8 | 7 | adantr 276 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} ⊆ 𝐵) |
9 | ensn1g 6796 | . . . 4 ⊢ (𝐴 ∈ 𝐵 → {𝐴} ≈ 1o) | |
10 | ensym 6780 | . . . 4 ⊢ (𝐵 ≈ 1o → 1o ≈ 𝐵) | |
11 | entr 6783 | . . . 4 ⊢ (({𝐴} ≈ 1o ∧ 1o ≈ 𝐵) → {𝐴} ≈ 𝐵) | |
12 | 9, 10, 11 | syl2an 289 | . . 3 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} ≈ 𝐵) |
13 | fisseneq 6930 | . . 3 ⊢ ((𝐵 ∈ Fin ∧ {𝐴} ⊆ 𝐵 ∧ {𝐴} ≈ 𝐵) → {𝐴} = 𝐵) | |
14 | 6, 8, 12, 13 | syl3anc 1238 | . 2 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → {𝐴} = 𝐵) |
15 | 14 | eqcomd 2183 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 = wceq 1353 ∈ wcel 2148 ⊆ wss 3129 {csn 3592 class class class wbr 4003 ωcom 4589 1oc1o 6409 ≈ cen 6737 Fincfn 6739 |
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-coll 4118 ax-sep 4121 ax-nul 4129 ax-pow 4174 ax-pr 4209 ax-un 4433 ax-setind 4536 ax-iinf 4587 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 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-ne 2348 df-ral 2460 df-rex 2461 df-reu 2462 df-rab 2464 df-v 2739 df-sbc 2963 df-csb 3058 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-if 3535 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-iun 3888 df-br 4004 df-opab 4065 df-mpt 4066 df-tr 4102 df-id 4293 df-iord 4366 df-on 4368 df-suc 4371 df-iom 4590 df-xp 4632 df-rel 4633 df-cnv 4634 df-co 4635 df-dm 4636 df-rn 4637 df-res 4638 df-ima 4639 df-iota 5178 df-fun 5218 df-fn 5219 df-f 5220 df-f1 5221 df-fo 5222 df-f1o 5223 df-fv 5224 df-1o 6416 df-er 6534 df-en 6740 df-fin 6742 |
This theorem is referenced by: en1eqsnbi 6947 1nsgtrivd 13072 en1top 13508 |
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