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Mathbox for Richard Penner |
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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 2733 | . . . 4 ⊢ ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴}) | |
2 | vex 3472 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | sneqr 4836 | . . . . 5 ⊢ ({𝑥} = {𝐴} → 𝑥 = 𝐴) |
4 | sneq 4633 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
5 | 3, 4 | impbii 208 | . . . 4 ⊢ ({𝑥} = {𝐴} ↔ 𝑥 = 𝐴) |
6 | 1, 5 | bitri 275 | . . 3 ⊢ ({𝐴} = {𝑥} ↔ 𝑥 = 𝐴) |
7 | 6 | exbii 1842 | . 2 ⊢ (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴) |
8 | en1 9020 | . 2 ⊢ ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥}) | |
9 | isset 3481 | . 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 205 = wceq 1533 ∃wex 1773 ∈ wcel 2098 Vcvv 3468 {csn 4623 class class class wbr 5141 1oc1o 8457 ≈ cen 8935 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2163 ax-ext 2697 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5142 df-opab 5204 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-suc 6363 df-iota 6488 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-1o 8464 df-en 8939 |
This theorem is referenced by: snen1el 42833 |
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