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Mirrors > Home > MPE Home > Th. List > card1 | Structured version Visualization version GIF version |
Description: A set has cardinality one iff it is a singleton. (Contributed by Mario Carneiro, 10-Jan-2013.) |
Ref | Expression |
---|---|
card1 | ⊢ ((card‘𝐴) = 1o ↔ ∃𝑥 𝐴 = {𝑥}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1onn 8267 | . . . . . . . 8 ⊢ 1o ∈ ω | |
2 | cardnn 9394 | . . . . . . . 8 ⊢ (1o ∈ ω → (card‘1o) = 1o) | |
3 | 1, 2 | ax-mp 5 | . . . . . . 7 ⊢ (card‘1o) = 1o |
4 | 1n0 8121 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
5 | 3, 4 | eqnetri 3088 | . . . . . 6 ⊢ (card‘1o) ≠ ∅ |
6 | carden2a 9397 | . . . . . 6 ⊢ (((card‘1o) = (card‘𝐴) ∧ (card‘1o) ≠ ∅) → 1o ≈ 𝐴) | |
7 | 5, 6 | mpan2 689 | . . . . 5 ⊢ ((card‘1o) = (card‘𝐴) → 1o ≈ 𝐴) |
8 | 7 | eqcoms 2831 | . . . 4 ⊢ ((card‘𝐴) = (card‘1o) → 1o ≈ 𝐴) |
9 | 8 | ensymd 8562 | . . 3 ⊢ ((card‘𝐴) = (card‘1o) → 𝐴 ≈ 1o) |
10 | carden2b 9398 | . . 3 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o)) | |
11 | 9, 10 | impbii 211 | . 2 ⊢ ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o) |
12 | 3 | eqeq2i 2836 | . 2 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o) |
13 | en1 8578 | . 2 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | |
14 | 11, 12, 13 | 3bitr3i 303 | 1 ⊢ ((card‘𝐴) = 1o ↔ ∃𝑥 𝐴 = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 208 = wceq 1537 ∃wex 1780 ∈ wcel 2114 ≠ wne 3018 ∅c0 4293 {csn 4569 class class class wbr 5068 ‘cfv 6357 ωcom 7582 1oc1o 8097 ≈ cen 8508 cardccrd 9366 |
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-3or 1084 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-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 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-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-om 7583 df-1o 8104 df-er 8291 df-en 8512 df-dom 8513 df-sdom 8514 df-fin 8515 df-card 9370 |
This theorem is referenced by: cardsn 9400 |
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