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Mirrors > Home > ILE Home > Th. List > unfiexmid | Unicode version |
Description: If the union of any two finite sets is finite, excluded middle follows. Remark 8.1.17 of [AczelRathjen], p. 74. (Contributed by Mario Carneiro and Jim Kingdon, 5-Mar-2022.) |
Ref | Expression |
---|---|
unfiexmid.1 |
Ref | Expression |
---|---|
unfiexmid |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-pr 3567 | . . . . 5 | |
2 | unfiexmid.1 | . . . . . . 7 | |
3 | 2 | rgen2a 2511 | . . . . . 6 |
4 | df1o2 6376 | . . . . . . . . . 10 | |
5 | rabeq 2704 | . . . . . . . . . 10 | |
6 | 4, 5 | ax-mp 5 | . . . . . . . . 9 |
7 | ordtriexmidlem 4478 | . . . . . . . . 9 | |
8 | 6, 7 | eqeltri 2230 | . . . . . . . 8 |
9 | snfig 6759 | . . . . . . . 8 | |
10 | 8, 9 | ax-mp 5 | . . . . . . 7 |
11 | 1onn 6467 | . . . . . . . 8 | |
12 | snfig 6759 | . . . . . . . 8 | |
13 | 11, 12 | ax-mp 5 | . . . . . . 7 |
14 | uneq1 3254 | . . . . . . . . 9 | |
15 | 14 | eleq1d 2226 | . . . . . . . 8 |
16 | uneq2 3255 | . . . . . . . . 9 | |
17 | 16 | eleq1d 2226 | . . . . . . . 8 |
18 | 15, 17 | rspc2v 2829 | . . . . . . 7 |
19 | 10, 13, 18 | mp2an 423 | . . . . . 6 |
20 | 3, 19 | ax-mp 5 | . . . . 5 |
21 | 1, 20 | eqeltri 2230 | . . . 4 |
22 | 8 | elexi 2724 | . . . . 5 |
23 | 22 | prid1 3665 | . . . 4 |
24 | 11 | elexi 2724 | . . . . 5 |
25 | 24 | prid2 3666 | . . . 4 |
26 | fidceq 6814 | . . . 4 DECID | |
27 | 21, 23, 25, 26 | mp3an 1319 | . . 3 DECID |
28 | exmiddc 822 | . . 3 DECID | |
29 | 27, 28 | ax-mp 5 | . 2 |
30 | 4 | eqeq2i 2168 | . . . 4 |
31 | 0ex 4091 | . . . . 5 | |
32 | biidd 171 | . . . . 5 | |
33 | 31, 32 | rabsnt 3634 | . . . 4 |
34 | 30, 33 | sylbi 120 | . . 3 |
35 | df-rab 2444 | . . . . 5 | |
36 | iba 298 | . . . . . 6 | |
37 | 36 | abbi2dv 2276 | . . . . 5 |
38 | 35, 37 | eqtr4id 2209 | . . . 4 |
39 | 38 | con3i 622 | . . 3 |
40 | 34, 39 | orim12i 749 | . 2 |
41 | 29, 40 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 698 DECID wdc 820 wceq 1335 wcel 2128 cab 2143 wral 2435 crab 2439 cun 3100 c0 3394 csn 3560 cpr 3561 con0 4323 com 4549 c1o 6356 cfn 6685 |
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 4082 ax-nul 4090 ax-pow 4135 ax-pr 4169 ax-un 4393 ax-setind 4496 ax-iinf 4547 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 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 3773 df-int 3808 df-br 3966 df-opab 4026 df-tr 4063 df-id 4253 df-iord 4326 df-on 4328 df-suc 4331 df-iom 4550 df-xp 4592 df-rel 4593 df-cnv 4594 df-co 4595 df-dm 4596 df-rn 4597 df-iota 5135 df-fun 5172 df-fn 5173 df-f 5174 df-f1 5175 df-fo 5176 df-f1o 5177 df-fv 5178 df-1o 6363 df-en 6686 df-fin 6688 |
This theorem is referenced by: (None) |
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