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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 3590 | . . . . 5 | |
2 | unfiexmid.1 | . . . . . . 7 | |
3 | 2 | rgen2a 2524 | . . . . . 6 |
4 | df1o2 6408 | . . . . . . . . . 10 | |
5 | rabeq 2722 | . . . . . . . . . 10 | |
6 | 4, 5 | ax-mp 5 | . . . . . . . . 9 |
7 | ordtriexmidlem 4503 | . . . . . . . . 9 | |
8 | 6, 7 | eqeltri 2243 | . . . . . . . 8 |
9 | snfig 6792 | . . . . . . . 8 | |
10 | 8, 9 | ax-mp 5 | . . . . . . 7 |
11 | 1onn 6499 | . . . . . . . 8 | |
12 | snfig 6792 | . . . . . . . 8 | |
13 | 11, 12 | ax-mp 5 | . . . . . . 7 |
14 | uneq1 3274 | . . . . . . . . 9 | |
15 | 14 | eleq1d 2239 | . . . . . . . 8 |
16 | uneq2 3275 | . . . . . . . . 9 | |
17 | 16 | eleq1d 2239 | . . . . . . . 8 |
18 | 15, 17 | rspc2v 2847 | . . . . . . 7 |
19 | 10, 13, 18 | mp2an 424 | . . . . . 6 |
20 | 3, 19 | ax-mp 5 | . . . . 5 |
21 | 1, 20 | eqeltri 2243 | . . . 4 |
22 | 8 | elexi 2742 | . . . . 5 |
23 | 22 | prid1 3689 | . . . 4 |
24 | 11 | elexi 2742 | . . . . 5 |
25 | 24 | prid2 3690 | . . . 4 |
26 | fidceq 6847 | . . . 4 DECID | |
27 | 21, 23, 25, 26 | mp3an 1332 | . . 3 DECID |
28 | exmiddc 831 | . . 3 DECID | |
29 | 27, 28 | ax-mp 5 | . 2 |
30 | 4 | eqeq2i 2181 | . . . 4 |
31 | 0ex 4116 | . . . . 5 | |
32 | biidd 171 | . . . . 5 | |
33 | 31, 32 | rabsnt 3658 | . . . 4 |
34 | 30, 33 | sylbi 120 | . . 3 |
35 | df-rab 2457 | . . . . 5 | |
36 | iba 298 | . . . . . 6 | |
37 | 36 | abbi2dv 2289 | . . . . 5 |
38 | 35, 37 | eqtr4id 2222 | . . . 4 |
39 | 38 | con3i 627 | . . 3 |
40 | 34, 39 | orim12i 754 | . 2 |
41 | 29, 40 | ax-mp 5 | 1 |
Colors of variables: wff set class |
Syntax hints: wn 3 wi 4 wa 103 wo 703 DECID wdc 829 wceq 1348 wcel 2141 cab 2156 wral 2448 crab 2452 cun 3119 c0 3414 csn 3583 cpr 3584 con0 4348 com 4574 c1o 6388 cfn 6718 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-f1 5203 df-fo 5204 df-f1o 5205 df-fv 5206 df-1o 6395 df-en 6719 df-fin 6721 |
This theorem is referenced by: (None) |
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