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| Mirrors > Home > MPE Home > Th. List > unir1 | Structured version Visualization version GIF version | ||
| Description: The cumulative hierarchy of sets covers the universe. Proposition 4.45 (b) to (a) of [Mendelson] p. 281. (Contributed by NM, 27-Sep-2004.) (Revised by Mario Carneiro, 8-Jun-2013.) | 
| Ref | Expression | 
|---|---|
| unir1 | ⊢ ∪ (𝑅1 “ On) = V | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | setind 9775 | . 2 ⊢ (∀𝑥(𝑥 ⊆ ∪ (𝑅1 “ On) → 𝑥 ∈ ∪ (𝑅1 “ On)) → ∪ (𝑅1 “ On) = V) | |
| 2 | vex 3483 | . . . 4 ⊢ 𝑥 ∈ V | |
| 3 | 2 | r1elss 9847 | . . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑥 ⊆ ∪ (𝑅1 “ On)) | 
| 4 | 3 | biimpri 228 | . 2 ⊢ (𝑥 ⊆ ∪ (𝑅1 “ On) → 𝑥 ∈ ∪ (𝑅1 “ On)) | 
| 5 | 1, 4 | mpg 1796 | 1 ⊢ ∪ (𝑅1 “ On) = V | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 = wceq 1539 ∈ wcel 2107 Vcvv 3479 ⊆ wss 3950 ∪ cuni 4906 “ cima 5687 Oncon0 6383 𝑅1cr1 9803 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 ax-reg 9633 ax-inf2 9682 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-om 7889 df-2nd 8016 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-r1 9805 | 
| This theorem is referenced by: jech9.3 9855 rankwflem 9856 rankval 9857 rankr1g 9873 rankid 9874 ssrankr1 9876 rankel 9880 rankval3 9881 rankpw 9884 rankss 9890 ranksn 9895 rankuni2 9896 rankun 9897 rankpr 9898 rankop 9899 r1rankid 9900 rankeq0 9902 rankr1b 9905 dfac12a 10190 hsmex2 10474 grutsk 10863 grurankcld 44257 | 
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