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Mirrors > Home > MPE Home > Th. List > oawordeu | Structured version Visualization version GIF version |
Description: Existence theorem for weak ordering of ordinal sum. Proposition 8.8 of [TakeutiZaring] p. 59. (Contributed by NM, 11-Dec-2004.) |
Ref | Expression |
---|---|
oawordeu | ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sseq1 3767 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 ⊆ 𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵)) | |
2 | oveq1 6820 | . . . . . 6 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (𝐴 +𝑜 𝑥) = (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥)) | |
3 | 2 | eqeq1d 2762 | . . . . 5 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ((𝐴 +𝑜 𝑥) = 𝐵 ↔ (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = 𝐵)) |
4 | 3 | reubidv 3265 | . . . 4 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → (∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵 ↔ ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = 𝐵)) |
5 | 1, 4 | imbi12d 333 | . . 3 ⊢ (𝐴 = if(𝐴 ∈ On, 𝐴, ∅) → ((𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵) ↔ (if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵 → ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = 𝐵))) |
6 | sseq2 3768 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → (if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵 ↔ if(𝐴 ∈ On, 𝐴, ∅) ⊆ if(𝐵 ∈ On, 𝐵, ∅))) | |
7 | eqeq2 2771 | . . . . 5 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → ((if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = 𝐵 ↔ (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = if(𝐵 ∈ On, 𝐵, ∅))) | |
8 | 7 | reubidv 3265 | . . . 4 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → (∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = 𝐵 ↔ ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = if(𝐵 ∈ On, 𝐵, ∅))) |
9 | 6, 8 | imbi12d 333 | . . 3 ⊢ (𝐵 = if(𝐵 ∈ On, 𝐵, ∅) → ((if(𝐴 ∈ On, 𝐴, ∅) ⊆ 𝐵 → ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = 𝐵) ↔ (if(𝐴 ∈ On, 𝐴, ∅) ⊆ if(𝐵 ∈ On, 𝐵, ∅) → ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = if(𝐵 ∈ On, 𝐵, ∅)))) |
10 | 0elon 5939 | . . . . 5 ⊢ ∅ ∈ On | |
11 | 10 | elimel 4294 | . . . 4 ⊢ if(𝐴 ∈ On, 𝐴, ∅) ∈ On |
12 | 10 | elimel 4294 | . . . 4 ⊢ if(𝐵 ∈ On, 𝐵, ∅) ∈ On |
13 | eqid 2760 | . . . 4 ⊢ {𝑦 ∈ On ∣ if(𝐵 ∈ On, 𝐵, ∅) ⊆ (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑦)} = {𝑦 ∈ On ∣ if(𝐵 ∈ On, 𝐵, ∅) ⊆ (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑦)} | |
14 | 11, 12, 13 | oawordeulem 7803 | . . 3 ⊢ (if(𝐴 ∈ On, 𝐴, ∅) ⊆ if(𝐵 ∈ On, 𝐵, ∅) → ∃!𝑥 ∈ On (if(𝐴 ∈ On, 𝐴, ∅) +𝑜 𝑥) = if(𝐵 ∈ On, 𝐵, ∅)) |
15 | 5, 9, 14 | dedth2h 4284 | . 2 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ⊆ 𝐵 → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵)) |
16 | 15 | imp 444 | 1 ⊢ (((𝐴 ∈ On ∧ 𝐵 ∈ On) ∧ 𝐴 ⊆ 𝐵) → ∃!𝑥 ∈ On (𝐴 +𝑜 𝑥) = 𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1632 ∈ wcel 2139 ∃!wreu 3052 {crab 3054 ⊆ wss 3715 ∅c0 4058 ifcif 4230 Oncon0 5884 (class class class)co 6813 +𝑜 coa 7726 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-8 2141 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 ax-rep 4923 ax-sep 4933 ax-nul 4941 ax-pow 4992 ax-pr 5055 ax-un 7114 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-eu 2611 df-mo 2612 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-ne 2933 df-ral 3055 df-rex 3056 df-reu 3057 df-rmo 3058 df-rab 3059 df-v 3342 df-sbc 3577 df-csb 3675 df-dif 3718 df-un 3720 df-in 3722 df-ss 3729 df-pss 3731 df-nul 4059 df-if 4231 df-pw 4304 df-sn 4322 df-pr 4324 df-tp 4326 df-op 4328 df-uni 4589 df-int 4628 df-iun 4674 df-br 4805 df-opab 4865 df-mpt 4882 df-tr 4905 df-id 5174 df-eprel 5179 df-po 5187 df-so 5188 df-fr 5225 df-we 5227 df-xp 5272 df-rel 5273 df-cnv 5274 df-co 5275 df-dm 5276 df-rn 5277 df-res 5278 df-ima 5279 df-pred 5841 df-ord 5887 df-on 5888 df-lim 5889 df-suc 5890 df-iota 6012 df-fun 6051 df-fn 6052 df-f 6053 df-f1 6054 df-fo 6055 df-f1o 6056 df-fv 6057 df-ov 6816 df-oprab 6817 df-mpt2 6818 df-om 7231 df-wrecs 7576 df-recs 7637 df-rdg 7675 df-oadd 7733 |
This theorem is referenced by: oawordex 7806 oaf1o 7812 oaabs 7893 oaabs2 7894 finxpreclem4 33542 |
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