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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 2742 | . . . 4 ⊢ ({𝐴} = {𝑥} ↔ {𝑥} = {𝐴}) | |
2 | vex 3482 | . . . . . 6 ⊢ 𝑥 ∈ V | |
3 | 2 | sneqr 4845 | . . . . 5 ⊢ ({𝑥} = {𝐴} → 𝑥 = 𝐴) |
4 | sneq 4641 | . . . . 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 1845 | . 2 ⊢ (∃𝑥{𝐴} = {𝑥} ↔ ∃𝑥 𝑥 = 𝐴) |
8 | en1 9063 | . 2 ⊢ ({𝐴} ≈ 1o ↔ ∃𝑥{𝐴} = {𝑥}) | |
9 | isset 3492 | . 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 1537 ∃wex 1776 ∈ wcel 2106 Vcvv 3478 {csn 4631 class class class wbr 5148 1oc1o 8498 ≈ cen 8981 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-1o 8505 df-en 8985 |
This theorem is referenced by: snen1el 43515 |
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