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Mirrors > Home > MPE Home > Th. List > n0ons | Structured version Visualization version GIF version |
Description: A surreal natural is a surreal ordinal. (Contributed by Scott Fenton, 2-Apr-2025.) |
Ref | Expression |
---|---|
n0ons | ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | n0sno 28343 | . . . . 5 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ No ) | |
2 | 1sno 27887 | . . . . 5 ⊢ 1s ∈ No | |
3 | subscl 28107 | . . . . 5 ⊢ ((𝐴 ∈ No ∧ 1s ∈ No ) → (𝐴 -s 1s ) ∈ No ) | |
4 | 1, 2, 3 | sylancl 586 | . . . 4 ⊢ (𝐴 ∈ ℕ0s → (𝐴 -s 1s ) ∈ No ) |
5 | ovex 7464 | . . . . 5 ⊢ (𝐴 -s 1s ) ∈ V | |
6 | 5 | snelpw 5456 | . . . 4 ⊢ ((𝐴 -s 1s ) ∈ No ↔ {(𝐴 -s 1s )} ∈ 𝒫 No ) |
7 | 4, 6 | sylib 218 | . . 3 ⊢ (𝐴 ∈ ℕ0s → {(𝐴 -s 1s )} ∈ 𝒫 No ) |
8 | n0scut 28353 | . . 3 ⊢ (𝐴 ∈ ℕ0s → 𝐴 = ({(𝐴 -s 1s )} |s ∅)) | |
9 | oveq1 7438 | . . . . 5 ⊢ (𝑥 = {(𝐴 -s 1s )} → (𝑥 |s ∅) = ({(𝐴 -s 1s )} |s ∅)) | |
10 | 9 | eqeq2d 2746 | . . . 4 ⊢ (𝑥 = {(𝐴 -s 1s )} → (𝐴 = (𝑥 |s ∅) ↔ 𝐴 = ({(𝐴 -s 1s )} |s ∅))) |
11 | 10 | rspcev 3622 | . . 3 ⊢ (({(𝐴 -s 1s )} ∈ 𝒫 No ∧ 𝐴 = ({(𝐴 -s 1s )} |s ∅)) → ∃𝑥 ∈ 𝒫 No 𝐴 = (𝑥 |s ∅)) |
12 | 7, 8, 11 | syl2anc 584 | . 2 ⊢ (𝐴 ∈ ℕ0s → ∃𝑥 ∈ 𝒫 No 𝐴 = (𝑥 |s ∅)) |
13 | elons2 28296 | . 2 ⊢ (𝐴 ∈ Ons ↔ ∃𝑥 ∈ 𝒫 No 𝐴 = (𝑥 |s ∅)) | |
14 | 12, 13 | sylibr 234 | 1 ⊢ (𝐴 ∈ ℕ0s → 𝐴 ∈ Ons) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ∃wrex 3068 ∅c0 4339 𝒫 cpw 4605 {csn 4631 (class class class)co 7431 No csur 27699 |s cscut 27842 1s c1s 27883 -s csubs 28067 Onscons 28289 ℕ0scnn0s 28333 |
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-rep 5285 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-rmo 3378 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 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-tp 4636 df-op 4638 df-ot 4640 df-uni 4913 df-int 4952 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-se 5642 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-pred 6323 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-riota 7388 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-2o 8506 df-nadd 8703 df-no 27702 df-slt 27703 df-bday 27704 df-sle 27805 df-sslt 27841 df-scut 27843 df-0s 27884 df-1s 27885 df-made 27901 df-old 27902 df-left 27904 df-right 27905 df-norec 27986 df-norec2 27997 df-adds 28008 df-negs 28068 df-subs 28069 df-ons 28290 df-n0s 28335 |
This theorem is referenced by: (None) |
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