| Mathbox for Thierry Arnoux |
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| Mirrors > Home > MPE Home > Th. List > Mathboxes > snct | Structured version Visualization version GIF version | ||
| Description: A singleton is countable. (Contributed by Thierry Arnoux, 16-Sep-2016.) |
| Ref | Expression |
|---|---|
| snct | ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≼ ω) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | ensn1g 9062 | . 2 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o) | |
| 2 | peano1 7910 | . . . . 5 ⊢ ∅ ∈ ω | |
| 3 | 2 | ne0ii 4344 | . . . 4 ⊢ ω ≠ ∅ |
| 4 | omex 9683 | . . . . 5 ⊢ ω ∈ V | |
| 5 | 4 | 0sdom 9147 | . . . 4 ⊢ (∅ ≺ ω ↔ ω ≠ ∅) |
| 6 | 3, 5 | mpbir 231 | . . 3 ⊢ ∅ ≺ ω |
| 7 | 0sdom1dom 9274 | . . 3 ⊢ (∅ ≺ ω ↔ 1o ≼ ω) | |
| 8 | 6, 7 | mpbi 230 | . 2 ⊢ 1o ≼ ω |
| 9 | endomtr 9052 | . 2 ⊢ (({𝐴} ≈ 1o ∧ 1o ≼ ω) → {𝐴} ≼ ω) | |
| 10 | 1, 8, 9 | sylancl 586 | 1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≼ ω) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 ≠ wne 2940 ∅c0 4333 {csn 4626 class class class wbr 5143 ωcom 7887 1oc1o 8499 ≈ cen 8982 ≼ cdom 8983 ≺ csdm 8984 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 ax-inf2 9681 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-om 7888 df-1o 8506 df-en 8986 df-dom 8987 df-sdom 8988 |
| This theorem is referenced by: prct 32726 oms0 34299 |
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