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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 2735 | . . . 4 ⊢ ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴}) | |
2 | vex 3475 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | sneqr 4842 | . . . . 5 ⊢ ({𝑥} = {𝐴} → 𝑥 = 𝐴) |
4 | sneq 4639 | . . . . 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 1843 | . 2 ⊢ (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴) |
8 | en1 9046 | . 2 ⊢ ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥}) | |
9 | isset 3484 | . 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 1534 ∃wex 1774 ∈ wcel 2099 Vcvv 3471 {csn 4629 class class class wbr 5148 1oc1o 8480 ≈ cen 8961 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-1o 8487 df-en 8965 |
This theorem is referenced by: snen1el 42955 |
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