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| Mirrors > Home > MPE Home > Th. List > en1b | Structured version Visualization version GIF version | ||
| Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by Mario Carneiro, 17-Jan-2015.) Avoid ax-un 7755. (Revised by BTernaryTau, 24-Sep-2024.) | 
| Ref | Expression | 
|---|---|
| en1b | ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | en1 9064 | . . 3 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | |
| 2 | id 22 | . . . . 5 ⊢ (𝐴 = {𝑥} → 𝐴 = {𝑥}) | |
| 3 | unieq 4918 | . . . . . . 7 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = ∪ {𝑥}) | |
| 4 | unisnv 4927 | . . . . . . 7 ⊢ ∪ {𝑥} = 𝑥 | |
| 5 | 3, 4 | eqtrdi 2793 | . . . . . 6 ⊢ (𝐴 = {𝑥} → ∪ 𝐴 = 𝑥) | 
| 6 | 5 | sneqd 4638 | . . . . 5 ⊢ (𝐴 = {𝑥} → {∪ 𝐴} = {𝑥}) | 
| 7 | 2, 6 | eqtr4d 2780 | . . . 4 ⊢ (𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) | 
| 8 | 7 | exlimiv 1930 | . . 3 ⊢ (∃𝑥 𝐴 = {𝑥} → 𝐴 = {∪ 𝐴}) | 
| 9 | 1, 8 | sylbi 217 | . 2 ⊢ (𝐴 ≈ 1o → 𝐴 = {∪ 𝐴}) | 
| 10 | id 22 | . . 3 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 = {∪ 𝐴}) | |
| 11 | eqsnuniex 5361 | . . . 4 ⊢ (𝐴 = {∪ 𝐴} → ∪ 𝐴 ∈ V) | |
| 12 | ensn1g 9062 | . . . 4 ⊢ (∪ 𝐴 ∈ V → {∪ 𝐴} ≈ 1o) | |
| 13 | 11, 12 | syl 17 | . . 3 ⊢ (𝐴 = {∪ 𝐴} → {∪ 𝐴} ≈ 1o) | 
| 14 | 10, 13 | eqbrtrd 5165 | . 2 ⊢ (𝐴 = {∪ 𝐴} → 𝐴 ≈ 1o) | 
| 15 | 9, 14 | impbii 209 | 1 ⊢ (𝐴 ≈ 1o ↔ 𝐴 = {∪ 𝐴}) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 {csn 4626 ∪ cuni 4907 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-11 2157 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: en1uniel 9069 sylow2alem2 19636 sylow2a 19637 frgpcyg 21592 ptcmplem3 24062 cnextfvval 24073 cnextcn 24075 minveclem4a 25464 isppw 27157 xrge0tsmsbi 33066 | 
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