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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 8648 | . 2 ⊢ (∀𝑥(𝑥 ⊆ ∪ (𝑅1 “ On) → 𝑥 ∈ ∪ (𝑅1 “ On)) → ∪ (𝑅1 “ On) = V) | |
2 | vex 3234 | . . . 4 ⊢ 𝑥 ∈ V | |
3 | 2 | r1elss 8707 | . . 3 ⊢ (𝑥 ∈ ∪ (𝑅1 “ On) ↔ 𝑥 ⊆ ∪ (𝑅1 “ On)) |
4 | 3 | biimpri 218 | . 2 ⊢ (𝑥 ⊆ ∪ (𝑅1 “ On) → 𝑥 ∈ ∪ (𝑅1 “ On)) |
5 | 1, 4 | mpg 1764 | 1 ⊢ ∪ (𝑅1 “ On) = V |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1523 ∈ wcel 2030 Vcvv 3231 ⊆ wss 3607 ∪ cuni 4468 “ cima 5146 Oncon0 5761 𝑅1cr1 8663 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1762 ax-4 1777 ax-5 1879 ax-6 1945 ax-7 1981 ax-8 2032 ax-9 2039 ax-10 2059 ax-11 2074 ax-12 2087 ax-13 2282 ax-ext 2631 ax-rep 4804 ax-sep 4814 ax-nul 4822 ax-pow 4873 ax-pr 4936 ax-un 6991 ax-reg 8538 ax-inf2 8576 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1055 df-3an 1056 df-tru 1526 df-ex 1745 df-nf 1750 df-sb 1938 df-eu 2502 df-mo 2503 df-clab 2638 df-cleq 2644 df-clel 2647 df-nfc 2782 df-ne 2824 df-ral 2946 df-rex 2947 df-reu 2948 df-rab 2950 df-v 3233 df-sbc 3469 df-csb 3567 df-dif 3610 df-un 3612 df-in 3614 df-ss 3621 df-pss 3623 df-nul 3949 df-if 4120 df-pw 4193 df-sn 4211 df-pr 4213 df-tp 4215 df-op 4217 df-uni 4469 df-int 4508 df-iun 4554 df-br 4686 df-opab 4746 df-mpt 4763 df-tr 4786 df-id 5053 df-eprel 5058 df-po 5064 df-so 5065 df-fr 5102 df-we 5104 df-xp 5149 df-rel 5150 df-cnv 5151 df-co 5152 df-dm 5153 df-rn 5154 df-res 5155 df-ima 5156 df-pred 5718 df-ord 5764 df-on 5765 df-lim 5766 df-suc 5767 df-iota 5889 df-fun 5928 df-fn 5929 df-f 5930 df-f1 5931 df-fo 5932 df-f1o 5933 df-fv 5934 df-om 7108 df-wrecs 7452 df-recs 7513 df-rdg 7551 df-r1 8665 |
This theorem is referenced by: jech9.3 8715 rankwflem 8716 rankval 8717 rankr1g 8733 rankid 8734 ssrankr1 8736 rankel 8740 rankval3 8741 rankpw 8744 rankss 8750 ranksn 8755 rankuni2 8756 rankun 8757 rankpr 8758 rankop 8759 r1rankid 8760 rankeq0 8762 rankr1b 8765 dfac12a 9008 hsmex2 9293 grutsk 9682 |
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