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Mirrors > Home > MPE Home > Th. List > rankxplim2 | Structured version Visualization version Unicode version |
Description: If the rank of a Cartesian product is a limit ordinal, so is the rank of the union of its arguments. (Contributed by NM, 19-Sep-2006.) |
Ref | Expression |
---|---|
rankxplim.1 |
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rankxplim.2 |
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Ref | Expression |
---|---|
rankxplim2 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ellim 5484 |
. . . 4
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2 | n0i 3735 |
. . . 4
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3 | 1, 2 | syl 17 |
. . 3
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4 | df-ne 2623 |
. . . 4
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5 | rankxplim.1 |
. . . . . . 7
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6 | rankxplim.2 |
. . . . . . 7
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7 | 5, 6 | xpex 6592 |
. . . . . 6
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8 | 7 | rankeq0 8329 |
. . . . 5
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9 | 8 | notbii 298 |
. . . 4
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10 | 4, 9 | bitr2i 254 |
. . 3
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11 | 3, 10 | sylib 200 |
. 2
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12 | limuni2 5483 |
. . . 4
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13 | limuni2 5483 |
. . . 4
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14 | 12, 13 | syl 17 |
. . 3
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15 | rankuni 8331 |
. . . . . 6
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16 | rankuni 8331 |
. . . . . . 7
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17 | 16 | unieqi 4206 |
. . . . . 6
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18 | 15, 17 | eqtr2i 2473 |
. . . . 5
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19 | unixp 5368 |
. . . . . 6
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20 | 19 | fveq2d 5867 |
. . . . 5
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21 | 18, 20 | syl5eq 2496 |
. . . 4
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22 | limeq 5434 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
23 | 21, 22 | syl 17 |
. . 3
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24 | 14, 23 | syl5ib 223 |
. 2
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25 | 11, 24 | mpcom 37 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1668 ax-4 1681 ax-5 1757 ax-6 1804 ax-7 1850 ax-8 1888 ax-9 1895 ax-10 1914 ax-11 1919 ax-12 1932 ax-13 2090 ax-ext 2430 ax-rep 4514 ax-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-reg 8104 ax-inf2 8143 |
This theorem depends on definitions: df-bi 189 df-or 372 df-an 373 df-3or 985 df-3an 986 df-tru 1446 df-ex 1663 df-nf 1667 df-sb 1797 df-eu 2302 df-mo 2303 df-clab 2437 df-cleq 2443 df-clel 2446 df-nfc 2580 df-ne 2623 df-ral 2741 df-rex 2742 df-reu 2743 df-rab 2745 df-v 3046 df-sbc 3267 df-csb 3363 df-dif 3406 df-un 3408 df-in 3410 df-ss 3417 df-pss 3419 df-nul 3731 df-if 3881 df-pw 3952 df-sn 3968 df-pr 3970 df-tp 3972 df-op 3974 df-uni 4198 df-int 4234 df-iun 4279 df-br 4402 df-opab 4461 df-mpt 4462 df-tr 4497 df-eprel 4744 df-id 4748 df-po 4754 df-so 4755 df-fr 4792 df-we 4794 df-xp 4839 df-rel 4840 df-cnv 4841 df-co 4842 df-dm 4843 df-rn 4844 df-res 4845 df-ima 4846 df-pred 5379 df-ord 5425 df-on 5426 df-lim 5427 df-suc 5428 df-iota 5545 df-fun 5583 df-fn 5584 df-f 5585 df-f1 5586 df-fo 5587 df-f1o 5588 df-fv 5589 df-om 6690 df-wrecs 7025 df-recs 7087 df-rdg 7125 df-r1 8232 df-rank 8233 |
This theorem is referenced by: rankxpsuc 8350 |
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