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| Mirrors > Home > MPE Home > Th. List > Mathboxes > sn1dom | Structured version Visualization version GIF version | ||
| Description: A singleton is dominated by ordinal one. (Contributed by RP, 29-Oct-2023.) | 
| Ref | Expression | 
|---|---|
| sn1dom | ⊢ {𝐴} ≼ 1o | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | ensn1g 9062 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o) | |
| 2 | 1on 8518 | . . . 4 ⊢ 1o ∈ On | |
| 3 | domrefg 9027 | . . . 4 ⊢ (1o ∈ On → 1o ≼ 1o) | |
| 4 | 2, 3 | ax-mp 5 | . . 3 ⊢ 1o ≼ 1o | 
| 5 | endomtr 9052 | . . 3 ⊢ (({𝐴} ≈ 1o ∧ 1o ≼ 1o) → {𝐴} ≼ 1o) | |
| 6 | 1, 4, 5 | sylancl 586 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ≼ 1o) | 
| 7 | snprc 4717 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
| 8 | snex 5436 | . . . . 5 ⊢ {𝐴} ∈ V | |
| 9 | eqeng 9026 | . . . . 5 ⊢ ({𝐴} ∈ V → ({𝐴} = ∅ → {𝐴} ≈ ∅)) | |
| 10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ({𝐴} = ∅ → {𝐴} ≈ ∅) | 
| 11 | 7, 10 | sylbi 217 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} ≈ ∅) | 
| 12 | 0domg 9140 | . . . 4 ⊢ (1o ∈ On → ∅ ≼ 1o) | |
| 13 | 2, 12 | ax-mp 5 | . . 3 ⊢ ∅ ≼ 1o | 
| 14 | endomtr 9052 | . . 3 ⊢ (({𝐴} ≈ ∅ ∧ ∅ ≼ 1o) → {𝐴} ≼ 1o) | |
| 15 | 11, 13, 14 | sylancl 586 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ≼ 1o) | 
| 16 | 6, 15 | pm2.61i 182 | 1 ⊢ {𝐴} ≼ 1o | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 ∅c0 4333 {csn 4626 class class class wbr 5143 Oncon0 6384 1oc1o 8499 ≈ cen 8982 ≼ cdom 8983 | 
| 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 | 
| 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-suc 6390 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-1o 8506 df-en 8986 df-dom 8987 | 
| This theorem is referenced by: pr2dom 43540 tr3dom 43541 | 
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