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Mirrors > Home > MPE Home > Th. List > hsmex | Structured version Visualization version GIF version |
Description: The collection of hereditarily size-limited well-founded sets comprise a set. The proof is that of Randall Holmes at http://math.boisestate.edu/~holmes/holmes/hereditary.pdf, with modifications to use Hartogs' theorem instead of the weak variant (inconsequentially weakening some intermediate results), and making the well-foundedness condition explicit to avoid a direct dependence on ax-reg 8664. (Contributed by Stefan O'Rear, 14-Feb-2015.) |
Ref | Expression |
---|---|
hsmex | ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | breq2 4808 | . . . . 5 ⊢ (𝑎 = 𝑋 → (𝑥 ≼ 𝑎 ↔ 𝑥 ≼ 𝑋)) | |
2 | 1 | ralbidv 3124 | . . . 4 ⊢ (𝑎 = 𝑋 → (∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎 ↔ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋)) |
3 | 2 | rabbidv 3329 | . . 3 ⊢ (𝑎 = 𝑋 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} = {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋}) |
4 | 3 | eleq1d 2824 | . 2 ⊢ (𝑎 = 𝑋 → ({𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} ∈ V ↔ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V)) |
5 | vex 3343 | . . 3 ⊢ 𝑎 ∈ V | |
6 | eqid 2760 | . . 3 ⊢ (rec((𝑑 ∈ V ↦ (har‘𝒫 (𝑎 × 𝑑))), (har‘𝒫 𝑎)) ↾ ω) = (rec((𝑑 ∈ V ↦ (har‘𝒫 (𝑎 × 𝑑))), (har‘𝒫 𝑎)) ↾ ω) | |
7 | rdgeq2 7678 | . . . . . 6 ⊢ (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) = rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑏)) | |
8 | unieq 4596 | . . . . . . . 8 ⊢ (𝑓 = 𝑐 → ∪ 𝑓 = ∪ 𝑐) | |
9 | 8 | cbvmptv 4902 | . . . . . . 7 ⊢ (𝑓 ∈ V ↦ ∪ 𝑓) = (𝑐 ∈ V ↦ ∪ 𝑐) |
10 | rdgeq1 7677 | . . . . . . 7 ⊢ ((𝑓 ∈ V ↦ ∪ 𝑓) = (𝑐 ∈ V ↦ ∪ 𝑐) → rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑏) = rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏)) | |
11 | 9, 10 | ax-mp 5 | . . . . . 6 ⊢ rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑏) = rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏) |
12 | 7, 11 | syl6eq 2810 | . . . . 5 ⊢ (𝑒 = 𝑏 → rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) = rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏)) |
13 | 12 | reseq1d 5550 | . . . 4 ⊢ (𝑒 = 𝑏 → (rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) ↾ ω) = (rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏) ↾ ω)) |
14 | 13 | cbvmptv 4902 | . . 3 ⊢ (𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) ↾ ω)) = (𝑏 ∈ V ↦ (rec((𝑐 ∈ V ↦ ∪ 𝑐), 𝑏) ↾ ω)) |
15 | eqid 2760 | . . 3 ⊢ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} = {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} | |
16 | eqid 2760 | . . 3 ⊢ OrdIso( E , (rank “ (((𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) ↾ ω))‘𝑧)‘𝑦))) = OrdIso( E , (rank “ (((𝑒 ∈ V ↦ (rec((𝑓 ∈ V ↦ ∪ 𝑓), 𝑒) ↾ ω))‘𝑧)‘𝑦))) | |
17 | 5, 6, 14, 15, 16 | hsmexlem6 9465 | . 2 ⊢ {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑎} ∈ V |
18 | 4, 17 | vtoclg 3406 | 1 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∈ ∪ (𝑅1 “ On) ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1632 ∈ wcel 2139 ∀wral 3050 {crab 3054 Vcvv 3340 𝒫 cpw 4302 {csn 4321 ∪ cuni 4588 class class class wbr 4804 ↦ cmpt 4881 E cep 5178 × cxp 5264 ↾ cres 5268 “ cima 5269 Oncon0 5884 ‘cfv 6049 ωcom 7231 reccrdg 7675 ≼ cdom 8121 OrdIsocoi 8581 harchar 8628 TCctc 8787 𝑅1cr1 8800 rankcrnk 8801 |
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 7115 ax-inf2 8713 |
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-se 5226 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-isom 6058 df-riota 6775 df-om 7232 df-1st 7334 df-2nd 7335 df-wrecs 7577 df-smo 7613 df-recs 7638 df-rdg 7676 df-er 7913 df-en 8124 df-dom 8125 df-sdom 8126 df-oi 8582 df-har 8630 df-wdom 8631 df-tc 8788 df-r1 8802 df-rank 8803 |
This theorem is referenced by: hsmex2 9467 |
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