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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 8574 | . 2 ⊢ (𝐴 ∈ 𝐶 → {𝐴} ≈ 1o) | |
2 | ensn1g 8574 | . . 3 ⊢ (𝐵 ∈ 𝐷 → {𝐵} ≈ 1o) | |
3 | 2 | ensymd 8560 | . 2 ⊢ (𝐵 ∈ 𝐷 → 1o ≈ {𝐵}) |
4 | entr 8561 | . 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 4567 class class class wbr 5066 1oc1o 8095 ≈ cen 8506 |
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 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
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 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-suc 6197 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-1o 8102 df-er 8289 df-en 8510 |
This theorem is referenced by: enpr2d 8597 difsnen 8599 domunsncan 8617 domunsn 8667 limensuci 8693 infensuc 8695 sucdom2 8714 dif1en 8751 dif1card 9436 fin23lem26 9747 unsnen 9975 canthp1lem1 10074 fzennn 13337 hashsng 13731 mreexexlem4d 16918 |
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