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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 5365, ax-un 7755. (Revised by BTernaryTau, 4-Jan-2025.) |
| Ref | Expression |
|---|---|
| en1eqsn | ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | en1 9064 | . . 3 ⊢ (𝐵 ≈ 1o ↔ ∃𝑥 𝐵 = {𝑥}) | |
| 2 | eleq2 2830 | . . . . . . . 8 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 ↔ 𝐴 ∈ {𝑥})) | |
| 3 | elsni 4643 | . . . . . . . . 9 ⊢ (𝐴 ∈ {𝑥} → 𝐴 = 𝑥) | |
| 4 | 3 | sneqd 4638 | . . . . . . . 8 ⊢ (𝐴 ∈ {𝑥} → {𝐴} = {𝑥}) |
| 5 | 2, 4 | biimtrdi 253 | . . . . . . 7 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → {𝐴} = {𝑥})) |
| 6 | 5 | imp 406 | . . . . . 6 ⊢ ((𝐵 = {𝑥} ∧ 𝐴 ∈ 𝐵) → {𝐴} = {𝑥}) |
| 7 | eqtr3 2763 | . . . . . 6 ⊢ ((𝐵 = {𝑥} ∧ {𝐴} = {𝑥}) → 𝐵 = {𝐴}) | |
| 8 | 6, 7 | syldan 591 | . . . . 5 ⊢ ((𝐵 = {𝑥} ∧ 𝐴 ∈ 𝐵) → 𝐵 = {𝐴}) |
| 9 | 8 | ex 412 | . . . 4 ⊢ (𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴})) |
| 10 | 9 | exlimiv 1930 | . . 3 ⊢ (∃𝑥 𝐵 = {𝑥} → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴})) |
| 11 | 1, 10 | sylbi 217 | . 2 ⊢ (𝐵 ≈ 1o → (𝐴 ∈ 𝐵 → 𝐵 = {𝐴})) |
| 12 | 11 | impcom 407 | 1 ⊢ ((𝐴 ∈ 𝐵 ∧ 𝐵 ≈ 1o) → 𝐵 = {𝐴}) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {csn 4626 class class class wbr 5143 1oc1o 8499 ≈ cen 8982 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-1o 8506 df-en 8986 |
| This theorem is referenced by: en1eqsnbi 9310 1nsgtrivd 19192 gex1 19609 0cyg 19911 pgpfac1lem3a 20096 pgpfaclem3 20103 0ring 20526 en1top 22991 cnextfres1 24076 xrge0tsmseq 33067 sconnpi1 35244 rngoueqz 37947 isdmn3 38081 |
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