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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 9061 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o) | |
2 | 1on 8517 | . . . 4 ⊢ 1o ∈ On | |
3 | domrefg 9026 | . . . 4 ⊢ (1o ∈ On → 1o ≼ 1o) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ 1o ≼ 1o |
5 | endomtr 9051 | . . 3 ⊢ (({𝐴} ≈ 1o ∧ 1o ≼ 1o) → {𝐴} ≼ 1o) | |
6 | 1, 4, 5 | sylancl 586 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ≼ 1o) |
7 | snprc 4722 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
8 | snex 5442 | . . . . 5 ⊢ {𝐴} ∈ V | |
9 | eqeng 9025 | . . . . 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 9139 | . . . 4 ⊢ (1o ∈ On → ∅ ≼ 1o) | |
13 | 2, 12 | ax-mp 5 | . . 3 ⊢ ∅ ≼ 1o |
14 | endomtr 9051 | . . 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 1537 ∈ wcel 2106 Vcvv 3478 ∅c0 4339 {csn 4631 class class class wbr 5148 Oncon0 6386 1oc1o 8498 ≈ cen 8981 ≼ cdom 8982 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-suc 6392 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-1o 8505 df-en 8985 df-dom 8986 |
This theorem is referenced by: pr2dom 43517 tr3dom 43518 |
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