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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 8660 | . . . . . . . 8 ⊢ 1o ∈ ω | |
2 | cardnn 9986 | . . . . . . . 8 ⊢ (1o ∈ ω → (card‘1o) = 1o) | |
3 | 1, 2 | ax-mp 5 | . . . . . . 7 ⊢ (card‘1o) = 1o |
4 | 1n0 8508 | . . . . . . 7 ⊢ 1o ≠ ∅ | |
5 | 3, 4 | eqnetri 3008 | . . . . . 6 ⊢ (card‘1o) ≠ ∅ |
6 | carden2a 9989 | . . . . . 6 ⊢ (((card‘1o) = (card‘𝐴) ∧ (card‘1o) ≠ ∅) → 1o ≈ 𝐴) | |
7 | 5, 6 | mpan2 690 | . . . . 5 ⊢ ((card‘1o) = (card‘𝐴) → 1o ≈ 𝐴) |
8 | 7 | eqcoms 2736 | . . . 4 ⊢ ((card‘𝐴) = (card‘1o) → 1o ≈ 𝐴) |
9 | 8 | ensymd 9025 | . . 3 ⊢ ((card‘𝐴) = (card‘1o) → 𝐴 ≈ 1o) |
10 | carden2b 9990 | . . 3 ⊢ (𝐴 ≈ 1o → (card‘𝐴) = (card‘1o)) | |
11 | 9, 10 | impbii 208 | . 2 ⊢ ((card‘𝐴) = (card‘1o) ↔ 𝐴 ≈ 1o) |
12 | 3 | eqeq2i 2741 | . 2 ⊢ ((card‘𝐴) = (card‘1o) ↔ (card‘𝐴) = 1o) |
13 | en1 9045 | . 2 ⊢ (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥}) | |
14 | 11, 12, 13 | 3bitr3i 301 | 1 ⊢ ((card‘𝐴) = 1o ↔ ∃𝑥 𝐴 = {𝑥}) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 = wceq 1534 ∃wex 1774 ∈ wcel 2099 ≠ wne 2937 ∅c0 4323 {csn 4629 class class class wbr 5148 ‘cfv 6548 ωcom 7870 1oc1o 8479 ≈ cen 8960 cardccrd 9958 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pow 5365 ax-pr 5429 ax-un 7740 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3374 df-rab 3430 df-v 3473 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-int 4950 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6372 df-on 6373 df-lim 6374 df-suc 6375 df-iota 6500 df-fun 6550 df-fn 6551 df-f 6552 df-f1 6553 df-fo 6554 df-f1o 6555 df-fv 6556 df-om 7871 df-1o 8486 df-er 8724 df-en 8964 df-dom 8965 df-sdom 8966 df-fin 8967 df-card 9962 |
This theorem is referenced by: cardsn 9992 |
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