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Mirrors > Home > MPE Home > Th. List > hsmex3 | Structured version Visualization version GIF version |
Description: The set of hereditary size-limited sets, assuming ax-reg 9056, using strict comparison (an easy corollary by separation). (Contributed by Stefan O'Rear, 11-Feb-2015.) |
Ref | Expression |
---|---|
hsmex3 | ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≺ 𝑋} ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sdomdom 8537 | . . . 4 ⊢ (𝑥 ≺ 𝑋 → 𝑥 ≼ 𝑋) | |
2 | 1 | ralimi 3160 | . . 3 ⊢ (∀𝑥 ∈ (TC‘{𝑠})𝑥 ≺ 𝑋 → ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋) |
3 | 2 | ss2abi 4043 | . 2 ⊢ {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≺ 𝑋} ⊆ {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} |
4 | hsmex2 9855 | . 2 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V) | |
5 | ssexg 5227 | . 2 ⊢ (({𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≺ 𝑋} ⊆ {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∧ {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≼ 𝑋} ∈ V) → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≺ 𝑋} ∈ V) | |
6 | 3, 4, 5 | sylancr 589 | 1 ⊢ (𝑋 ∈ 𝑉 → {𝑠 ∣ ∀𝑥 ∈ (TC‘{𝑠})𝑥 ≺ 𝑋} ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2114 {cab 2799 ∀wral 3138 Vcvv 3494 ⊆ wss 3936 {csn 4567 class class class wbr 5066 ‘cfv 6355 ≼ cdom 8507 ≺ csdm 8508 TCctc 9178 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 ax-reg 9056 ax-inf2 9104 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-int 4877 df-iun 4921 df-iin 4922 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-se 5515 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-isom 6364 df-riota 7114 df-om 7581 df-1st 7689 df-2nd 7690 df-wrecs 7947 df-smo 7983 df-recs 8008 df-rdg 8046 df-er 8289 df-en 8510 df-dom 8511 df-sdom 8512 df-oi 8974 df-har 9022 df-wdom 9023 df-tc 9179 df-r1 9193 df-rank 9194 |
This theorem is referenced by: (None) |
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