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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.) Avoid ax-pow 5325, ax-un 7677. (Revised by BTernaryTau, 4-Jan-2025.) |
Ref | Expression |
---|---|
en1eqsn | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | en1 8972 | . . 3 ⊢ (𝐵 ≈ 1o ↔ ∃𝑥 𝐵 = {𝑥}) | |
2 | eleq2 2827 | . . . . . . . 8 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ {𝑥})) | |
3 | elsni 4608 | . . . . . . . . 9 ⊢ (𝐴 ∈ {𝑥} → 𝐴 = 𝑥) | |
4 | 3 | sneqd 4603 | . . . . . . . 8 ⊢ (𝐴 ∈ {𝑥} → {𝐴} = {𝑥}) |
5 | 2, 4 | syl6bi 253 | . . . . . . 7 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → {𝐴} = {𝑥})) |
6 | 5 | imp 408 | . . . . . 6 ⊢ ((𝐵 = {𝑥} ∧ 𝐴 ∈ 𝐵) → {𝐴} = {𝑥}) |
7 | eqtr3 2763 | . . . . . 6 ⊢ ((𝐵 = {𝑥} ∧ {𝐴} = {𝑥}) → 𝐵 = {𝐴}) | |
8 | 6, 7 | syldan 592 | . . . . 5 ⊢ ((𝐵 = {𝑥} ∧ 𝐴 ∈ 𝐵) → 𝐵 = {𝐴}) |
9 | 8 | ex 414 | . . . 4 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴})) |
10 | 9 | exlimiv 1934 | . . 3 ⊢ (∃𝑥 𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴})) |
11 | 1, 10 | sylbi 216 | . 2 ⊢ (𝐵 ≈ 1o → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴})) |
12 | 11 | impcom 409 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∃wex 1782 ∈ wcel 2107 {csn 4591 class class class wbr 5110 1oc1o 8410 ≈ cen 8887 |
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-10 2138 ax-12 2172 ax-ext 2708 ax-sep 5261 ax-nul 5268 ax-pr 5389 |
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-nf 1787 df-sb 2069 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2815 df-ne 2945 df-ral 3066 df-rex 3075 df-reu 3357 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-opab 5173 df-id 5536 df-xp 5644 df-rel 5645 df-cnv 5646 df-co 5647 df-dm 5648 df-rn 5649 df-res 5650 df-ima 5651 df-suc 6328 df-iota 6453 df-fun 6503 df-fn 6504 df-f 6505 df-f1 6506 df-fo 6507 df-f1o 6508 df-fv 6509 df-1o 8417 df-en 8891 |
This theorem is referenced by: en1eqsnbi 9227 1nsgtrivd 18983 gex1 19380 0cyg 19677 pgpfac1lem3a 19862 pgpfaclem3 19869 0ring 20756 en1top 22350 cnextfres1 23435 xrge0tsmseq 31943 sconnpi1 33873 rngoueqz 36428 isdmn3 36562 |
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