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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 8557 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o) | |
2 | 1on 8092 | . . . 4 ⊢ 1o ∈ On | |
3 | domrefg 8527 | . . . 4 ⊢ (1o ∈ On → 1o ≼ 1o) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ 1o ≼ 1o |
5 | endomtr 8550 | . . 3 ⊢ (({𝐴} ≈ 1o ∧ 1o ≼ 1o) → {𝐴} ≼ 1o) | |
6 | 1, 4, 5 | sylancl 589 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ≼ 1o) |
7 | snprc 4613 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
8 | snex 5297 | . . . . 5 ⊢ {𝐴} ∈ V | |
9 | eqeng 8526 | . . . . 5 ⊢ ({𝐴} ∈ V → ({𝐴} = ∅ → {𝐴} ≈ ∅)) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ({𝐴} = ∅ → {𝐴} ≈ ∅) |
11 | 7, 10 | sylbi 220 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} ≈ ∅) |
12 | 0domg 8628 | . . . 4 ⊢ (1o ∈ On → ∅ ≼ 1o) | |
13 | 2, 12 | ax-mp 5 | . . 3 ⊢ ∅ ≼ 1o |
14 | endomtr 8550 | . . 3 ⊢ (({𝐴} ≈ ∅ ∧ ∅ ≼ 1o) → {𝐴} ≼ 1o) | |
15 | 11, 13, 14 | sylancl 589 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ≼ 1o) |
16 | 6, 15 | pm2.61i 185 | 1 ⊢ {𝐴} ≼ 1o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 {csn 4525 class class class wbr 5030 Oncon0 6159 1oc1o 8078 ≈ cen 8489 ≼ cdom 8490 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-ord 6162 df-on 6163 df-suc 6165 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-1o 8085 df-en 8493 df-dom 8494 |
This theorem is referenced by: pr2dom 40235 tr3dom 40236 |
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