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Mirrors > Home > ILE Home > Th. List > nndomo | Unicode version |
Description: Cardinal ordering agrees with natural number ordering. Example 3 of [Enderton] p. 146. (Contributed by NM, 17-Jun-1998.) |
Ref | Expression |
---|---|
nndomo |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | php5dom 6821 | . . . . . . . 8 | |
2 | 1 | ad2antlr 481 | . . . . . . 7 |
3 | domtr 6743 | . . . . . . . . 9 | |
4 | 3 | expcom 115 | . . . . . . . 8 |
5 | 4 | adantl 275 | . . . . . . 7 |
6 | 2, 5 | mtod 653 | . . . . . 6 |
7 | ssdomg 6736 | . . . . . . 7 | |
8 | 7 | ad2antrr 480 | . . . . . 6 |
9 | 6, 8 | mtod 653 | . . . . 5 |
10 | nnord 4584 | . . . . . . 7 | |
11 | ordsucss 4476 | . . . . . . 7 | |
12 | 10, 11 | syl 14 | . . . . . 6 |
13 | 12 | ad2antrr 480 | . . . . 5 |
14 | 9, 13 | mtod 653 | . . . 4 |
15 | nntri1 6456 | . . . . 5 | |
16 | 15 | adantr 274 | . . . 4 |
17 | 14, 16 | mpbird 166 | . . 3 |
18 | 17 | ex 114 | . 2 |
19 | ssdomg 6736 | . . 3 | |
20 | 19 | adantl 275 | . 2 |
21 | 18, 20 | impbid 128 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wb 104 wcel 2135 wss 3112 class class class wbr 3977 word 4335 csuc 4338 com 4562 cdom 6697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 604 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-13 2137 ax-14 2138 ax-ext 2146 ax-sep 4095 ax-nul 4103 ax-pow 4148 ax-pr 4182 ax-un 4406 ax-setind 4509 ax-iinf 4560 |
This theorem depends on definitions: df-bi 116 df-dc 825 df-3or 968 df-3an 969 df-tru 1345 df-fal 1348 df-nf 1448 df-sb 1750 df-eu 2016 df-mo 2017 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ne 2335 df-ral 2447 df-rex 2448 df-rab 2451 df-v 2724 df-sbc 2948 df-dif 3114 df-un 3116 df-in 3118 df-ss 3125 df-nul 3406 df-pw 3556 df-sn 3577 df-pr 3578 df-op 3580 df-uni 3785 df-int 3820 df-br 3978 df-opab 4039 df-tr 4076 df-id 4266 df-iord 4339 df-on 4341 df-suc 4344 df-iom 4563 df-xp 4605 df-rel 4606 df-cnv 4607 df-co 4608 df-dm 4609 df-rn 4610 df-res 4611 df-ima 4612 df-iota 5148 df-fun 5185 df-fn 5186 df-f 5187 df-f1 5188 df-fo 5189 df-f1o 5190 df-fv 5191 df-er 6493 df-en 6699 df-dom 6700 |
This theorem is referenced by: fisbth 6841 fientri3 6872 hashennnuni 10682 fihashdom 10706 pwf1oexmid 13741 |
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