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Mirrors > Home > MPE Home > Th. List > wfgru | Structured version Visualization version GIF version |
Description: The wellfounded part of a universe is another universe. (Contributed by Mario Carneiro, 17-Jun-2013.) |
Ref | Expression |
---|---|
wfgru | ⊢ (𝑈 ∈ Univ → (𝑈 ∩ ∪ (𝑅1 “ On)) ∈ Univ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dftr3 5178 | . . 3 ⊢ (Tr ∪ (𝑅1 “ On) ↔ ∀𝑥 ∈ ∪ (𝑅1 “ On)𝑥 ⊆ ∪ (𝑅1 “ On)) | |
2 | r1elssi 9236 | . . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝑥 ⊆ ∪ (𝑅1 “ On)) | |
3 | 1, 2 | mprgbir 3155 | . 2 ⊢ Tr ∪ (𝑅1 “ On) |
4 | pwwf 9238 | . . . . 5 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝒫 𝑥 ∈ ∪ (𝑅1 “ On)) | |
5 | 4 | biimpi 218 | . . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → 𝒫 𝑥 ∈ ∪ (𝑅1 “ On)) |
6 | prwf 9242 | . . . . 5 ⊢ ((𝑥 ∈ ∪ (𝑅1 “ On) ∧ 𝑦 ∈ ∪ (𝑅1 “ On)) → {𝑥, 𝑦} ∈ ∪ (𝑅1 “ On)) | |
7 | 6 | ralrimiva 3184 | . . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → ∀𝑦 ∈ ∪ (𝑅1 “ On){𝑥, 𝑦} ∈ ∪ (𝑅1 “ On)) |
8 | frn 6522 | . . . . . . 7 ⊢ (𝑦:𝑥⟶∪ (𝑅1 “ On) → ran 𝑦 ⊆ ∪ (𝑅1 “ On)) | |
9 | vex 3499 | . . . . . . . . . 10 ⊢ 𝑦 ∈ V | |
10 | 9 | rnex 7619 | . . . . . . . . 9 ⊢ ran 𝑦 ∈ V |
11 | 10 | r1elss 9237 | . . . . . . . 8 ⊢ (ran 𝑦 ∈ ∪ (𝑅1 “ On) ↔ ran 𝑦 ⊆ ∪ (𝑅1 “ On)) |
12 | uniwf 9250 | . . . . . . . 8 ⊢ (ran 𝑦 ∈ ∪ (𝑅1 “ On) ↔ ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On)) | |
13 | 11, 12 | bitr3i 279 | . . . . . . 7 ⊢ (ran 𝑦 ⊆ ∪ (𝑅1 “ On) ↔ ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On)) |
14 | 8, 13 | sylib 220 | . . . . . 6 ⊢ (𝑦:𝑥⟶∪ (𝑅1 “ On) → ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On)) |
15 | 14 | ax-gen 1796 | . . . . 5 ⊢ ∀𝑦(𝑦:𝑥⟶∪ (𝑅1 “ On) → ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On)) |
16 | 15 | a1i 11 | . . . 4 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → ∀𝑦(𝑦:𝑥⟶∪ (𝑅1 “ On) → ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On))) |
17 | 5, 7, 16 | 3jca 1124 | . . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) → (𝒫 𝑥 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ ∪ (𝑅1 “ On){𝑥, 𝑦} ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦(𝑦:𝑥⟶∪ (𝑅1 “ On) → ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On)))) |
18 | 17 | rgen 3150 | . 2 ⊢ ∀𝑥 ∈ ∪ (𝑅1 “ On)(𝒫 𝑥 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ ∪ (𝑅1 “ On){𝑥, 𝑦} ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦(𝑦:𝑥⟶∪ (𝑅1 “ On) → ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On))) |
19 | ingru 10239 | . 2 ⊢ ((Tr ∪ (𝑅1 “ On) ∧ ∀𝑥 ∈ ∪ (𝑅1 “ On)(𝒫 𝑥 ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦 ∈ ∪ (𝑅1 “ On){𝑥, 𝑦} ∈ ∪ (𝑅1 “ On) ∧ ∀𝑦(𝑦:𝑥⟶∪ (𝑅1 “ On) → ∪ ran 𝑦 ∈ ∪ (𝑅1 “ On)))) → (𝑈 ∈ Univ → (𝑈 ∩ ∪ (𝑅1 “ On)) ∈ Univ)) | |
20 | 3, 18, 19 | mp2an 690 | 1 ⊢ (𝑈 ∈ Univ → (𝑈 ∩ ∪ (𝑅1 “ On)) ∈ Univ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ w3a 1083 ∀wal 1535 ∈ wcel 2114 ∀wral 3140 ∩ cin 3937 ⊆ wss 3938 𝒫 cpw 4541 {cpr 4571 ∪ cuni 4840 Tr wtr 5174 ran crn 5558 “ cima 5560 Oncon0 6193 ⟶wf 6353 𝑅1cr1 9193 Univcgru 10214 |
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 2795 ax-rep 5192 ax-sep 5205 ax-nul 5212 ax-pow 5268 ax-pr 5332 ax-un 7463 |
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 2802 df-cleq 2816 df-clel 2895 df-nfc 2965 df-ne 3019 df-ral 3145 df-rex 3146 df-reu 3147 df-rab 3149 df-v 3498 df-sbc 3775 df-csb 3886 df-dif 3941 df-un 3943 df-in 3945 df-ss 3954 df-pss 3956 df-nul 4294 df-if 4470 df-pw 4543 df-sn 4570 df-pr 4572 df-tp 4574 df-op 4576 df-uni 4841 df-int 4879 df-iun 4923 df-br 5069 df-opab 5131 df-mpt 5149 df-tr 5175 df-id 5462 df-eprel 5467 df-po 5476 df-so 5477 df-fr 5516 df-we 5518 df-xp 5563 df-rel 5564 df-cnv 5565 df-co 5566 df-dm 5567 df-rn 5568 df-res 5569 df-ima 5570 df-pred 6150 df-ord 6196 df-on 6197 df-lim 6198 df-suc 6199 df-iota 6316 df-fun 6359 df-fn 6360 df-f 6361 df-f1 6362 df-fo 6363 df-f1o 6364 df-fv 6365 df-ov 7161 df-oprab 7162 df-mpo 7163 df-om 7583 df-1st 7691 df-2nd 7692 df-wrecs 7949 df-recs 8010 df-rdg 8048 df-map 8410 df-r1 9195 df-rank 9196 df-gru 10215 |
This theorem is referenced by: (None) |
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