![]() |
Mathbox for Richard Penner |
< Previous
Next >
Nearby theorems |
|
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 2739 | . . . 4 ⊢ ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴}) | |
2 | vex 3478 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | sneqr 4840 | . . . . 5 ⊢ ({𝑥} = {𝐴} → 𝑥 = 𝐴) |
4 | sneq 4637 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
5 | 3, 4 | impbii 208 | . . . 4 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴) |
6 | 1, 5 | bitri 274 | . . 3 ⊢ ({𝐴} = {𝑥} ↔ 𝑥 = 𝐴) |
7 | 6 | exbii 1850 | . 2 ⊢ (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴) |
8 | en1 9017 | . 2 ⊢ ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥}) | |
9 | isset 3487 | . 2 ⊢ (𝐴 ∈ V ↔ ∃𝑥 𝑥 = 𝐴) | |
10 | 7, 8, 9 | 3bitr4i 302 | 1 ⊢ ({𝐴} ≈ 1o ↔ 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1541 ∃wex 1781 ∈ wcel 2106 Vcvv 3474 {csn 4627 class class class wbr 5147 1oc1o 8455 ≈ cen 8932 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-12 2171 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-suc 6367 df-iota 6492 df-fun 6542 df-fn 6543 df-f 6544 df-f1 6545 df-fo 6546 df-f1o 6547 df-fv 6548 df-1o 8462 df-en 8936 |
This theorem is referenced by: snen1el 42261 |
Copyright terms: Public domain | W3C validator |