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Mirrors > Home > MPE Home > Th. List > Mathboxes > noetalem1 | Structured version Visualization version GIF version |
Description: Lemma for noeta 31868. Establish that our final surreal really is a surreal. (Contributed by Scott Fenton, 6-Dec-2021.) |
Ref | Expression |
---|---|
noetalem.1 | ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) |
noetalem.2 | ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜})) |
Ref | Expression |
---|---|
noetalem1 | ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 ∈ No ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | noetalem.2 | . 2 ⊢ 𝑍 = (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜})) | |
2 | noetalem.1 | . . . . 5 ⊢ 𝑆 = if(∃𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦, ((℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦) ∪ {〈dom (℩𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ¬ 𝑥 <s 𝑦), 2𝑜〉}), (𝑔 ∈ {𝑦 ∣ ∃𝑢 ∈ 𝐴 (𝑦 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑦) = (𝑣 ↾ suc 𝑦)))} ↦ (℩𝑥∃𝑢 ∈ 𝐴 (𝑔 ∈ dom 𝑢 ∧ ∀𝑣 ∈ 𝐴 (¬ 𝑣 <s 𝑢 → (𝑢 ↾ suc 𝑔) = (𝑣 ↾ suc 𝑔)) ∧ (𝑢‘𝑔) = 𝑥)))) | |
3 | 2 | nosupno 31849 | . . . 4 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V) → 𝑆 ∈ No ) |
4 | 3 | 3adant3 1081 | . . 3 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑆 ∈ No ) |
5 | bdayimaon 31843 | . . . 4 ⊢ (𝐵 ∈ V → suc ∪ ( bday “ 𝐵) ∈ On) | |
6 | 5 | 3ad2ant3 1084 | . . 3 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → suc ∪ ( bday “ 𝐵) ∈ On) |
7 | 1on 7567 | . . . . . 6 ⊢ 1𝑜 ∈ On | |
8 | 7 | elexi 3213 | . . . . 5 ⊢ 1𝑜 ∈ V |
9 | 8 | prid1 4297 | . . . 4 ⊢ 1𝑜 ∈ {1𝑜, 2𝑜} |
10 | 9 | noextendseq 31820 | . . 3 ⊢ ((𝑆 ∈ No ∧ suc ∪ ( bday “ 𝐵) ∈ On) → (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜})) ∈ No ) |
11 | 4, 6, 10 | syl2anc 693 | . 2 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → (𝑆 ∪ ((suc ∪ ( bday “ 𝐵) ∖ dom 𝑆) × {1𝑜})) ∈ No ) |
12 | 1, 11 | syl5eqel 2705 | 1 ⊢ ((𝐴 ⊆ No ∧ 𝐴 ∈ V ∧ 𝐵 ∈ V) → 𝑍 ∈ No ) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 384 ∧ w3a 1037 = wceq 1483 ∈ wcel 1990 {cab 2608 ∀wral 2912 ∃wrex 2913 Vcvv 3200 ∖ cdif 3571 ∪ cun 3572 ⊆ wss 3574 ifcif 4086 {csn 4177 〈cop 4183 ∪ cuni 4436 class class class wbr 4653 ↦ cmpt 4729 × cxp 5112 dom cdm 5114 ↾ cres 5116 “ cima 5117 Oncon0 5723 suc csuc 5725 ℩cio 5849 ‘cfv 5888 ℩crio 6610 1𝑜c1o 7553 2𝑜c2o 7554 No csur 31793 <s cslt 31794 bday cbday 31795 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-ord 5726 df-on 5727 df-suc 5729 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-riota 6611 df-1o 7560 df-2o 7561 df-no 31796 df-slt 31797 df-bday 31798 |
This theorem is referenced by: noetalem2 31864 noetalem3 31865 noetalem4 31866 noetalem5 31867 |
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