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Mirrors > Home > MPE Home > Th. List > inf3lem6 | Structured version Visualization version Unicode version |
Description: Lemma for our Axiom of Infinity => standard Axiom of Infinity. See inf3 8137 for detailed description. (Contributed by NM, 29-Oct-1996.) |
Ref | Expression |
---|---|
inf3lem.1 |
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inf3lem.2 |
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inf3lem.3 |
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inf3lem.4 |
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Ref | Expression |
---|---|
inf3lem6 |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | inf3lem.1 |
. . . . . . . . . . 11
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2 | inf3lem.2 |
. . . . . . . . . . 11
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3 | vex 3047 |
. . . . . . . . . . 11
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4 | vex 3047 |
. . . . . . . . . . 11
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5 | 1, 2, 3, 4 | inf3lem5 8134 |
. . . . . . . . . 10
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6 | dfpss2 3517 |
. . . . . . . . . . 11
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7 | 6 | simprbi 466 |
. . . . . . . . . 10
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8 | 5, 7 | syl6 34 |
. . . . . . . . 9
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9 | 8 | expdimp 439 |
. . . . . . . 8
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10 | 9 | adantrl 721 |
. . . . . . 7
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11 | 1, 2, 4, 3 | inf3lem5 8134 |
. . . . . . . . . 10
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
12 | dfpss2 3517 |
. . . . . . . . . . . 12
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13 | 12 | simprbi 466 |
. . . . . . . . . . 11
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14 | eqcom 2457 |
. . . . . . . . . . 11
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15 | 13, 14 | sylnib 306 |
. . . . . . . . . 10
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16 | 11, 15 | syl6 34 |
. . . . . . . . 9
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17 | 16 | expdimp 439 |
. . . . . . . 8
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18 | 17 | adantrr 722 |
. . . . . . 7
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19 | 10, 18 | jaod 382 |
. . . . . 6
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20 | 19 | con2d 119 |
. . . . 5
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21 | nnord 6697 |
. . . . . . 7
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22 | nnord 6697 |
. . . . . . 7
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23 | ordtri3 5458 |
. . . . . . 7
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24 | 21, 22, 23 | syl2an 480 |
. . . . . 6
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25 | 24 | adantl 468 |
. . . . 5
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26 | 20, 25 | sylibrd 238 |
. . . 4
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27 | 26 | ralrimivva 2808 |
. . 3
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28 | frfnom 7149 |
. . . . . 6
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29 | fneq1 5662 |
. . . . . 6
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30 | 28, 29 | mpbiri 237 |
. . . . 5
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31 | fvelrnb 5910 |
. . . . . . . 8
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32 | inf3lem.4 |
. . . . . . . . . . . 12
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33 | 1, 2, 4, 32 | inf3lemd 8129 |
. . . . . . . . . . 11
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34 | fvex 5873 |
. . . . . . . . . . . 12
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35 | 34 | elpw 3956 |
. . . . . . . . . . 11
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36 | 33, 35 | sylibr 216 |
. . . . . . . . . 10
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37 | eleq1 2516 |
. . . . . . . . . 10
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38 | 36, 37 | syl5ibcom 224 |
. . . . . . . . 9
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39 | 38 | rexlimiv 2872 |
. . . . . . . 8
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40 | 31, 39 | syl6bi 232 |
. . . . . . 7
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41 | 40 | ssrdv 3437 |
. . . . . 6
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42 | 41 | ancli 554 |
. . . . 5
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43 | 2, 30, 42 | mp2b 10 |
. . . 4
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44 | df-f 5585 |
. . . 4
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
45 | 43, 44 | mpbir 213 |
. . 3
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46 | 27, 45 | jctil 540 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
47 | dff13 6157 |
. 2
![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() | |
48 | 46, 47 | sylibr 216 |
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-sep 4524 ax-nul 4533 ax-pow 4580 ax-pr 4638 ax-un 6580 ax-reg 8104 |
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-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 |
This theorem is referenced by: inf3lem7 8136 dominf 8872 dominfac 8995 |
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