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Mirrors > Home > MPE Home > Th. List > en2sn | Structured version Visualization version GIF version |
Description: Two singletons are equinumerous. (Contributed by NM, 9-Nov-2003.) |
Ref | Expression |
---|---|
en2sn | ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ensn1g 7907 | . 2 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1𝑜) | |
2 | ensn1g 7907 | . . 3 ⊢ (𝐵 ∈ 𝐷 → {𝐵} ≈ 1𝑜) | |
3 | 2 | ensymd 7893 | . 2 ⊢ (𝐵 ∈ 𝐷 → 1𝑜 ≈ {𝐵}) |
4 | entr 7894 | . 2 ⊢ (({𝐴} ≈ 1𝑜 ∧ 1𝑜 ≈ {𝐵}) → {𝐴} ≈ {𝐵}) | |
5 | 1, 3, 4 | syl2an 493 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 ∈ 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-pow 4769 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-pw 4110 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-res 5050 df-ima 5051 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-er 7629 df-en 7842 |
This theorem is referenced by: difsnen 7927 domunsncan 7945 domunsn 7995 limensuci 8021 infensuc 8023 sucdom2 8041 dif1en 8078 dif1card 8716 fin23lem26 9030 unsnen 9254 canthp1lem1 9353 fzennn 12629 hashsng 13020 mreexexlem4d 16130 |
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