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Mirrors > Home > MPE Home > Th. List > ensn1g | Structured version Visualization version GIF version |
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.) |
Ref | Expression |
---|---|
ensn1g | ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1𝑜) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sneq 4135 | . . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
2 | 1 | breq1d 4593 | . 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1𝑜 ↔ {𝐴} ≈ 1𝑜)) |
3 | vex 3176 | . . 3 ⊢ 𝑥 ∈ V | |
4 | 3 | ensn1 7906 | . 2 ⊢ {𝑥} ≈ 1𝑜 |
5 | 2, 4 | vtoclg 3239 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1𝑜) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1475 ∈ wcel 1977 {csn 4125 class class class wbr 4583 1𝑜c1o 7440 ≈ cen 7838 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-suc 5646 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-1o 7447 df-en 7842 |
This theorem is referenced by: enpr1g 7908 en1b 7910 en2sn 7922 snfi 7923 snnen2o 8034 sucxpdom 8054 en1eqsn 8075 en1eqsnbi 8076 pr2nelem 8710 prdom2 8712 cda1en 8880 snct 28874 rngoueqz 32909 |
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