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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 8576 | . 2 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
2 | ensn1g 8576 | . . 3 ⊢ (𝐵 ∈ 𝐷 → {𝐵} ≈ 1o) | |
3 | 2 | ensymd 8562 | . 2 ⊢ (𝐵 ∈ 𝐷 → 1o ≈ {𝐵}) |
4 | entr 8563 | . 2 ⊢ (({𝐴} ≈ 1o ∧ 1o ≈ {𝐵}) → {𝐴} ≈ {𝐵}) | |
5 | 1, 3, 4 | syl2an 597 | 1 ⊢ ((𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐷) → {𝐴} ≈ {𝐵}) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 {csn 4569 class class class wbr 5068 1oc1o 8097 ≈ cen 8508 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2795 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ral 3145 df-rex 3146 df-rab 3149 df-v 3498 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-op 4576 df-uni 4841 df-br 5069 df-opab 5131 df-id 5462 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-suc 6199 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-1o 8104 df-er 8291 df-en 8512 |
This theorem is referenced by: enpr2d 8599 difsnen 8601 domunsncan 8619 domunsn 8669 limensuci 8695 infensuc 8697 sucdom2 8716 dif1en 8753 dif1card 9438 fin23lem26 9749 unsnen 9977 canthp1lem1 10076 fzennn 13339 hashsng 13733 mreexexlem4d 16920 |
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