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Mirrors > Home > MPE Home > Th. List > cardsn | Structured version Visualization version GIF version |
Description: A singleton has cardinality one. (Contributed by Mario Carneiro, 10-Jan-2013.) |
Ref | Expression |
---|---|
cardsn | ⊢ (𝐴 ∈ 𝑉 → (card‘{𝐴}) = 1o) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqid 2797 | . . 3 ⊢ {𝐴} = {𝐴} | |
2 | sneq 4488 | . . . . 5 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴}) | |
3 | 2 | eqeq2d 2807 | . . . 4 ⊢ (𝑥 = 𝐴 → ({𝐴} = {𝑥} ↔ {𝐴} = {𝐴})) |
4 | 3 | spcegv 3542 | . . 3 ⊢ (𝐴 ∈ 𝑉 → ({𝐴} = {𝐴} → ∃𝑥{𝐴} = {𝑥})) |
5 | 1, 4 | mpi 20 | . 2 ⊢ (𝐴 ∈ 𝑉 → ∃𝑥{𝐴} = {𝑥}) |
6 | card1 9250 | . 2 ⊢ ((card‘{𝐴}) = 1o ↔ ∃𝑥{𝐴} = {𝑥}) | |
7 | 5, 6 | sylibr 235 | 1 ⊢ (𝐴 ∈ 𝑉 → (card‘{𝐴}) = 1o) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1525 ∃wex 1765 ∈ wcel 2083 {csn 4478 ‘cfv 6232 1oc1o 7953 cardccrd 9217 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1781 ax-4 1795 ax-5 1892 ax-6 1951 ax-7 1996 ax-8 2085 ax-9 2093 ax-10 2114 ax-11 2128 ax-12 2143 ax-13 2346 ax-ext 2771 ax-sep 5101 ax-nul 5108 ax-pow 5164 ax-pr 5228 ax-un 7326 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1528 df-ex 1766 df-nf 1770 df-sb 2045 df-mo 2578 df-eu 2614 df-clab 2778 df-cleq 2790 df-clel 2865 df-nfc 2937 df-ne 2987 df-ral 3112 df-rex 3113 df-reu 3114 df-rab 3116 df-v 3442 df-sbc 3712 df-dif 3868 df-un 3870 df-in 3872 df-ss 3880 df-pss 3882 df-nul 4218 df-if 4388 df-pw 4461 df-sn 4479 df-pr 4481 df-tp 4483 df-op 4485 df-uni 4752 df-int 4789 df-br 4969 df-opab 5031 df-mpt 5048 df-tr 5071 df-id 5355 df-eprel 5360 df-po 5369 df-so 5370 df-fr 5409 df-we 5411 df-xp 5456 df-rel 5457 df-cnv 5458 df-co 5459 df-dm 5460 df-rn 5461 df-res 5462 df-ima 5463 df-ord 6076 df-on 6077 df-lim 6078 df-suc 6079 df-iota 6196 df-fun 6234 df-fn 6235 df-f 6236 df-f1 6237 df-fo 6238 df-f1o 6239 df-fv 6240 df-om 7444 df-1o 7960 df-er 8146 df-en 8365 df-dom 8366 df-sdom 8367 df-fin 8368 df-card 9221 |
This theorem is referenced by: ackbij1lem14 9508 cfsuc 9532 |
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