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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 8844 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o) | |
2 | 1on 8340 | . . . 4 ⊢ 1o ∈ On | |
3 | domrefg 8808 | . . . 4 ⊢ (1o ∈ On → 1o ≼ 1o) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ 1o ≼ 1o |
5 | endomtr 8833 | . . 3 ⊢ (({𝐴} ≈ 1o ∧ 1o ≼ 1o) → {𝐴} ≼ 1o) | |
6 | 1, 4, 5 | sylancl 587 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ≼ 1o) |
7 | snprc 4657 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
8 | snex 5363 | . . . . 5 ⊢ {𝐴} ∈ V | |
9 | eqeng 8807 | . . . . 5 ⊢ ({𝐴} ∈ V → ({𝐴} = ∅ → {𝐴} ≈ ∅)) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ({𝐴} = ∅ → {𝐴} ≈ ∅) |
11 | 7, 10 | sylbi 216 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} ≈ ∅) |
12 | 0domg 8925 | . . . 4 ⊢ (1o ∈ On → ∅ ≼ 1o) | |
13 | 2, 12 | ax-mp 5 | . . 3 ⊢ ∅ ≼ 1o |
14 | endomtr 8833 | . . 3 ⊢ (({𝐴} ≈ ∅ ∧ ∅ ≼ 1o) → {𝐴} ≼ 1o) | |
15 | 11, 13, 14 | sylancl 587 | . 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 1539 ∈ wcel 2104 Vcvv 3437 ∅c0 4262 {csn 4565 class class class wbr 5081 Oncon0 6281 1oc1o 8321 ≈ cen 8761 ≼ cdom 8762 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7620 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-rab 3287 df-v 3439 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-br 5082 df-opab 5144 df-tr 5199 df-id 5500 df-eprel 5506 df-po 5514 df-so 5515 df-fr 5555 df-we 5557 df-xp 5606 df-rel 5607 df-cnv 5608 df-co 5609 df-dm 5610 df-rn 5611 df-res 5612 df-ima 5613 df-ord 6284 df-on 6285 df-suc 6287 df-fun 6460 df-fn 6461 df-f 6462 df-f1 6463 df-fo 6464 df-f1o 6465 df-1o 8328 df-en 8765 df-dom 8766 |
This theorem is referenced by: pr2dom 41172 tr3dom 41173 |
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