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Theorem r1limwun 10680
Description: Each limit stage in the cumulative hierarchy is a weak universe. (Contributed by Mario Carneiro, 2-Jan-2017.)
Assertion
Ref Expression
r1limwun ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) β†’ (𝑅1β€˜π΄) ∈ WUni)

Proof of Theorem r1limwun
Dummy variables π‘₯ 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 r1tr 9720 . . 3 Tr (𝑅1β€˜π΄)
21a1i 11 . 2 ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) β†’ Tr (𝑅1β€˜π΄))
3 limelon 6385 . . . . . 6 ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) β†’ 𝐴 ∈ On)
4 r1fnon 9711 . . . . . . 7 𝑅1 Fn On
54fndmi 6610 . . . . . 6 dom 𝑅1 = On
63, 5eleqtrrdi 2845 . . . . 5 ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) β†’ 𝐴 ∈ dom 𝑅1)
7 onssr1 9775 . . . . 5 (𝐴 ∈ dom 𝑅1 β†’ 𝐴 βŠ† (𝑅1β€˜π΄))
86, 7syl 17 . . . 4 ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) β†’ 𝐴 βŠ† (𝑅1β€˜π΄))
9 0ellim 6384 . . . . 5 (Lim 𝐴 β†’ βˆ… ∈ 𝐴)
109adantl 483 . . . 4 ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) β†’ βˆ… ∈ 𝐴)
118, 10sseldd 3949 . . 3 ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) β†’ βˆ… ∈ (𝑅1β€˜π΄))
1211ne0d 4299 . 2 ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) β†’ (𝑅1β€˜π΄) β‰  βˆ…)
13 rankuni 9807 . . . . . 6 (rankβ€˜βˆͺ π‘₯) = βˆͺ (rankβ€˜π‘₯)
14 rankon 9739 . . . . . . . . 9 (rankβ€˜π‘₯) ∈ On
15 eloni 6331 . . . . . . . . 9 ((rankβ€˜π‘₯) ∈ On β†’ Ord (rankβ€˜π‘₯))
16 orduniss 6418 . . . . . . . . 9 (Ord (rankβ€˜π‘₯) β†’ βˆͺ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘₯))
1714, 15, 16mp2b 10 . . . . . . . 8 βˆͺ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘₯)
1817a1i 11 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ βˆͺ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘₯))
19 rankr1ai 9742 . . . . . . . 8 (π‘₯ ∈ (𝑅1β€˜π΄) β†’ (rankβ€˜π‘₯) ∈ 𝐴)
2019adantl 483 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ (rankβ€˜π‘₯) ∈ 𝐴)
21 onuni 7727 . . . . . . . . 9 ((rankβ€˜π‘₯) ∈ On β†’ βˆͺ (rankβ€˜π‘₯) ∈ On)
2214, 21ax-mp 5 . . . . . . . 8 βˆͺ (rankβ€˜π‘₯) ∈ On
233adantr 482 . . . . . . . 8 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ 𝐴 ∈ On)
24 ontr2 6368 . . . . . . . 8 ((βˆͺ (rankβ€˜π‘₯) ∈ On ∧ 𝐴 ∈ On) β†’ ((βˆͺ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘₯) ∧ (rankβ€˜π‘₯) ∈ 𝐴) β†’ βˆͺ (rankβ€˜π‘₯) ∈ 𝐴))
2522, 23, 24sylancr 588 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ ((βˆͺ (rankβ€˜π‘₯) βŠ† (rankβ€˜π‘₯) ∧ (rankβ€˜π‘₯) ∈ 𝐴) β†’ βˆͺ (rankβ€˜π‘₯) ∈ 𝐴))
2618, 20, 25mp2and 698 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ βˆͺ (rankβ€˜π‘₯) ∈ 𝐴)
2713, 26eqeltrid 2838 . . . . 5 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ (rankβ€˜βˆͺ π‘₯) ∈ 𝐴)
28 r1elwf 9740 . . . . . . . 8 (π‘₯ ∈ (𝑅1β€˜π΄) β†’ π‘₯ ∈ βˆͺ (𝑅1 β€œ On))
2928adantl 483 . . . . . . 7 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ π‘₯ ∈ βˆͺ (𝑅1 β€œ On))
30 uniwf 9763 . . . . . . 7 (π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ↔ βˆͺ π‘₯ ∈ βˆͺ (𝑅1 β€œ On))
3129, 30sylib 217 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ βˆͺ π‘₯ ∈ βˆͺ (𝑅1 β€œ On))
326adantr 482 . . . . . 6 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ 𝐴 ∈ dom 𝑅1)
33 rankr1ag 9746 . . . . . 6 ((βˆͺ π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐴 ∈ dom 𝑅1) β†’ (βˆͺ π‘₯ ∈ (𝑅1β€˜π΄) ↔ (rankβ€˜βˆͺ π‘₯) ∈ 𝐴))
3431, 32, 33syl2anc 585 . . . . 5 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ (βˆͺ π‘₯ ∈ (𝑅1β€˜π΄) ↔ (rankβ€˜βˆͺ π‘₯) ∈ 𝐴))
3527, 34mpbird 257 . . . 4 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ βˆͺ π‘₯ ∈ (𝑅1β€˜π΄))
36 r1pwcl 9791 . . . . . 6 (Lim 𝐴 β†’ (π‘₯ ∈ (𝑅1β€˜π΄) ↔ 𝒫 π‘₯ ∈ (𝑅1β€˜π΄)))
3736adantl 483 . . . . 5 ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) β†’ (π‘₯ ∈ (𝑅1β€˜π΄) ↔ 𝒫 π‘₯ ∈ (𝑅1β€˜π΄)))
3837biimpa 478 . . . 4 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ 𝒫 π‘₯ ∈ (𝑅1β€˜π΄))
3928ad2antlr 726 . . . . . . . 8 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ π‘₯ ∈ βˆͺ (𝑅1 β€œ On))
40 r1elwf 9740 . . . . . . . . 9 (𝑦 ∈ (𝑅1β€˜π΄) β†’ 𝑦 ∈ βˆͺ (𝑅1 β€œ On))
4140adantl 483 . . . . . . . 8 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ 𝑦 ∈ βˆͺ (𝑅1 β€œ On))
42 rankprb 9795 . . . . . . . 8 ((π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝑦 ∈ βˆͺ (𝑅1 β€œ On)) β†’ (rankβ€˜{π‘₯, 𝑦}) = suc ((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦)))
4339, 41, 42syl2anc 585 . . . . . . 7 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ (rankβ€˜{π‘₯, 𝑦}) = suc ((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦)))
44 limord 6381 . . . . . . . . . 10 (Lim 𝐴 β†’ Ord 𝐴)
4544ad3antlr 730 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ Ord 𝐴)
4620adantr 482 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ (rankβ€˜π‘₯) ∈ 𝐴)
47 rankr1ai 9742 . . . . . . . . . 10 (𝑦 ∈ (𝑅1β€˜π΄) β†’ (rankβ€˜π‘¦) ∈ 𝐴)
4847adantl 483 . . . . . . . . 9 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ (rankβ€˜π‘¦) ∈ 𝐴)
49 ordunel 7766 . . . . . . . . 9 ((Ord 𝐴 ∧ (rankβ€˜π‘₯) ∈ 𝐴 ∧ (rankβ€˜π‘¦) ∈ 𝐴) β†’ ((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦)) ∈ 𝐴)
5045, 46, 48, 49syl3anc 1372 . . . . . . . 8 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ ((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦)) ∈ 𝐴)
51 limsuc 7789 . . . . . . . . 9 (Lim 𝐴 β†’ (((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦)) ∈ 𝐴 ↔ suc ((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦)) ∈ 𝐴))
5251ad3antlr 730 . . . . . . . 8 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ (((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦)) ∈ 𝐴 ↔ suc ((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦)) ∈ 𝐴))
5350, 52mpbid 231 . . . . . . 7 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ suc ((rankβ€˜π‘₯) βˆͺ (rankβ€˜π‘¦)) ∈ 𝐴)
5443, 53eqeltrd 2834 . . . . . 6 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ (rankβ€˜{π‘₯, 𝑦}) ∈ 𝐴)
55 prwf 9755 . . . . . . . 8 ((π‘₯ ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝑦 ∈ βˆͺ (𝑅1 β€œ On)) β†’ {π‘₯, 𝑦} ∈ βˆͺ (𝑅1 β€œ On))
5639, 41, 55syl2anc 585 . . . . . . 7 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ {π‘₯, 𝑦} ∈ βˆͺ (𝑅1 β€œ On))
5732adantr 482 . . . . . . 7 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ 𝐴 ∈ dom 𝑅1)
58 rankr1ag 9746 . . . . . . 7 (({π‘₯, 𝑦} ∈ βˆͺ (𝑅1 β€œ On) ∧ 𝐴 ∈ dom 𝑅1) β†’ ({π‘₯, 𝑦} ∈ (𝑅1β€˜π΄) ↔ (rankβ€˜{π‘₯, 𝑦}) ∈ 𝐴))
5956, 57, 58syl2anc 585 . . . . . 6 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ ({π‘₯, 𝑦} ∈ (𝑅1β€˜π΄) ↔ (rankβ€˜{π‘₯, 𝑦}) ∈ 𝐴))
6054, 59mpbird 257 . . . . 5 ((((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) ∧ 𝑦 ∈ (𝑅1β€˜π΄)) β†’ {π‘₯, 𝑦} ∈ (𝑅1β€˜π΄))
6160ralrimiva 3140 . . . 4 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ βˆ€π‘¦ ∈ (𝑅1β€˜π΄){π‘₯, 𝑦} ∈ (𝑅1β€˜π΄))
6235, 38, 613jca 1129 . . 3 (((𝐴 ∈ 𝑉 ∧ Lim 𝐴) ∧ π‘₯ ∈ (𝑅1β€˜π΄)) β†’ (βˆͺ π‘₯ ∈ (𝑅1β€˜π΄) ∧ 𝒫 π‘₯ ∈ (𝑅1β€˜π΄) ∧ βˆ€π‘¦ ∈ (𝑅1β€˜π΄){π‘₯, 𝑦} ∈ (𝑅1β€˜π΄)))
6362ralrimiva 3140 . 2 ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) β†’ βˆ€π‘₯ ∈ (𝑅1β€˜π΄)(βˆͺ π‘₯ ∈ (𝑅1β€˜π΄) ∧ 𝒫 π‘₯ ∈ (𝑅1β€˜π΄) ∧ βˆ€π‘¦ ∈ (𝑅1β€˜π΄){π‘₯, 𝑦} ∈ (𝑅1β€˜π΄)))
64 fvex 6859 . . 3 (𝑅1β€˜π΄) ∈ V
65 iswun 10648 . . 3 ((𝑅1β€˜π΄) ∈ V β†’ ((𝑅1β€˜π΄) ∈ WUni ↔ (Tr (𝑅1β€˜π΄) ∧ (𝑅1β€˜π΄) β‰  βˆ… ∧ βˆ€π‘₯ ∈ (𝑅1β€˜π΄)(βˆͺ π‘₯ ∈ (𝑅1β€˜π΄) ∧ 𝒫 π‘₯ ∈ (𝑅1β€˜π΄) ∧ βˆ€π‘¦ ∈ (𝑅1β€˜π΄){π‘₯, 𝑦} ∈ (𝑅1β€˜π΄)))))
6664, 65ax-mp 5 . 2 ((𝑅1β€˜π΄) ∈ WUni ↔ (Tr (𝑅1β€˜π΄) ∧ (𝑅1β€˜π΄) β‰  βˆ… ∧ βˆ€π‘₯ ∈ (𝑅1β€˜π΄)(βˆͺ π‘₯ ∈ (𝑅1β€˜π΄) ∧ 𝒫 π‘₯ ∈ (𝑅1β€˜π΄) ∧ βˆ€π‘¦ ∈ (𝑅1β€˜π΄){π‘₯, 𝑦} ∈ (𝑅1β€˜π΄))))
672, 12, 63, 66syl3anbrc 1344 1 ((𝐴 ∈ 𝑉 ∧ Lim 𝐴) β†’ (𝑅1β€˜π΄) ∈ WUni)
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107   β‰  wne 2940  βˆ€wral 3061  Vcvv 3447   βˆͺ cun 3912   βŠ† wss 3914  βˆ…c0 4286  π’« cpw 4564  {cpr 4592  βˆͺ cuni 4869  Tr wtr 5226  dom cdm 5637   β€œ cima 5640  Ord word 6320  Oncon0 6321  Lim wlim 6322  suc csuc 6323  β€˜cfv 6500  π‘…1cr1 9706  rankcrnk 9707  WUnicwun 10644
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pow 5324  ax-pr 5388  ax-un 7676  ax-reg 9536  ax-inf2 9585
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-int 4912  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-r1 9708  df-rank 9709  df-wun 10646
This theorem is referenced by:  r1wunlim  10681  wunex3  10685
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