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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 8567 | . . 3 ⊢ (𝐴 ∈ V → {𝐴} ≈ 1o) | |
2 | 1on 8102 | . . . 4 ⊢ 1o ∈ On | |
3 | domrefg 8537 | . . . 4 ⊢ (1o ∈ On → 1o ≼ 1o) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ 1o ≼ 1o |
5 | endomtr 8560 | . . 3 ⊢ (({𝐴} ≈ 1o ∧ 1o ≼ 1o) → {𝐴} ≼ 1o) | |
6 | 1, 4, 5 | sylancl 588 | . 2 ⊢ (𝐴 ∈ V → {𝐴} ≼ 1o) |
7 | snprc 4646 | . . . 4 ⊢ (¬ 𝐴 ∈ V ↔ {𝐴} = ∅) | |
8 | snex 5325 | . . . . 5 ⊢ {𝐴} ∈ V | |
9 | eqeng 8536 | . . . . 5 ⊢ ({𝐴} ∈ V → ({𝐴} = ∅ → {𝐴} ≈ ∅)) | |
10 | 8, 9 | ax-mp 5 | . . . 4 ⊢ ({𝐴} = ∅ → {𝐴} ≈ ∅) |
11 | 7, 10 | sylbi 219 | . . 3 ⊢ (¬ 𝐴 ∈ V → {𝐴} ≈ ∅) |
12 | 0domg 8637 | . . . 4 ⊢ (1o ∈ On → ∅ ≼ 1o) | |
13 | 2, 12 | ax-mp 5 | . . 3 ⊢ ∅ ≼ 1o |
14 | endomtr 8560 | . . 3 ⊢ (({𝐴} ≈ ∅ ∧ ∅ ≼ 1o) → {𝐴} ≼ 1o) | |
15 | 11, 13, 14 | sylancl 588 | . 2 ⊢ (¬ 𝐴 ∈ V → {𝐴} ≼ 1o) |
16 | 6, 15 | pm2.61i 184 | 1 ⊢ {𝐴} ≼ 1o |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1536 ∈ wcel 2113 Vcvv 3491 ∅c0 4284 {csn 4560 class class class wbr 5059 Oncon0 6184 1oc1o 8088 ≈ cen 8499 ≼ cdom 8500 |
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 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2792 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5323 ax-un 7454 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1083 df-3an 1084 df-tru 1539 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2799 df-cleq 2813 df-clel 2892 df-nfc 2962 df-ne 3016 df-ral 3142 df-rex 3143 df-rab 3146 df-v 3493 df-sbc 3769 df-dif 3932 df-un 3934 df-in 3936 df-ss 3945 df-pss 3947 df-nul 4285 df-if 4461 df-pw 4534 df-sn 4561 df-pr 4563 df-tp 4565 df-op 4567 df-uni 4832 df-br 5060 df-opab 5122 df-tr 5166 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-ord 6187 df-on 6188 df-suc 6190 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-1o 8095 df-en 8503 df-dom 8504 |
This theorem is referenced by: pr2dom 39967 tr3dom 39968 |
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