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| Mirrors > Home > MPE Home > Th. List > Mathboxes > snen1g | Structured version Visualization version GIF version | ||
| Description: A singleton is equinumerous to ordinal one iff its content is a set. (Contributed by RP, 8-Oct-2023.) | 
| Ref | Expression | 
|---|---|
| snen1g | ⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqcom 2744 | . . . 4 ⊢ ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴}) | |
| 2 | vex 3484 | . . . . . 6 ⊢ 𝑥 ∈ V | |
| 3 | 2 | sneqr 4840 | . . . . 5 ⊢ ({𝑥} = {𝐴} → 𝑥 = 𝐴) | 
| 4 | sneq 4636 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
| 5 | 3, 4 | impbii 209 | . . . 4 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴) | 
| 6 | 1, 5 | bitri 275 | . . 3 ⊢ ({𝐴} = {𝑥} ↔ 𝑥 = 𝐴) | 
| 7 | 6 | exbii 1848 | . 2 ⊢ (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴) | 
| 8 | en1 9064 | . 2 ⊢ ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥}) | |
| 9 | isset 3494 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
| 10 | 7, 8, 9 | 3bitr4i 303 | 1 ⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 = wceq 1540 ∃wex 1779 ∈ wcel 2108 Vcvv 3480 {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: snen1el 43538 | 
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