Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > ILE Home > Th. List > infnninfOLD | Unicode version |
Description: Obsolete version of infnninf 7068 as of 10-Aug-2024. (Contributed by Jim Kingdon, 14-Jul-2022.) (Proof modification is discouraged.) (New usage is discouraged.) |
Ref | Expression |
---|---|
infnninfOLD | ℕ∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 1lt2o 6390 | . . . 4 | |
2 | 1 | fconst6 5370 | . . 3 |
3 | 2onn 6469 | . . . . 5 | |
4 | 3 | elexi 2724 | . . . 4 |
5 | omex 4553 | . . . 4 | |
6 | 4, 5 | elmap 6623 | . . 3 |
7 | 2, 6 | mpbir 145 | . 2 |
8 | peano2 4555 | . . . . . 6 | |
9 | 1oex 6372 | . . . . . . 7 | |
10 | 9 | fvconst2 5684 | . . . . . 6 |
11 | 8, 10 | syl 14 | . . . . 5 |
12 | 9 | fvconst2 5684 | . . . . 5 |
13 | 11, 12 | eqtr4d 2193 | . . . 4 |
14 | eqimss 3182 | . . . 4 | |
15 | 13, 14 | syl 14 | . . 3 |
16 | 15 | rgen 2510 | . 2 |
17 | fveq1 5468 | . . . . 5 | |
18 | fveq1 5468 | . . . . 5 | |
19 | 17, 18 | sseq12d 3159 | . . . 4 |
20 | 19 | ralbidv 2457 | . . 3 |
21 | df-nninf 7065 | . . 3 ℕ∞ | |
22 | 20, 21 | elrab2 2871 | . 2 ℕ∞ |
23 | 7, 16, 22 | mpbir2an 927 | 1 ℕ∞ |
Colors of variables: wff set class |
Syntax hints: wceq 1335 wcel 2128 wral 2435 wss 3102 csn 3560 csuc 4326 com 4550 cxp 4585 wf 5167 cfv 5171 (class class class)co 5825 c1o 6357 c2o 6358 cmap 6594 ℕ∞xnninf 7064 |
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 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4083 ax-nul 4091 ax-pow 4136 ax-pr 4170 ax-un 4394 ax-setind 4497 ax-iinf 4548 |
This theorem depends on definitions: df-bi 116 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3395 df-pw 3545 df-sn 3566 df-pr 3567 df-op 3569 df-uni 3774 df-int 3809 df-br 3967 df-opab 4027 df-mpt 4028 df-tr 4064 df-id 4254 df-iord 4327 df-on 4329 df-suc 4332 df-iom 4551 df-xp 4593 df-rel 4594 df-cnv 4595 df-co 4596 df-dm 4597 df-rn 4598 df-iota 5136 df-fun 5173 df-fn 5174 df-f 5175 df-fv 5179 df-ov 5828 df-oprab 5829 df-mpo 5830 df-1o 6364 df-2o 6365 df-map 6596 df-nninf 7065 |
This theorem is referenced by: fxnn0nninf 10341 |
Copyright terms: Public domain | W3C validator |