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Mirrors > Home > ILE Home > Th. List > inffiexmid | GIF version |
Description: If any given set is either finite or infinite, excluded middle follows. (Contributed by Jim Kingdon, 15-Jun-2022.) |
Ref | Expression |
---|---|
inffiexmid.1 | ⊢ (𝑥 ∈ Fin ∨ ω ≼ 𝑥) |
Ref | Expression |
---|---|
inffiexmid | ⊢ (𝜑 ∨ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | omex 4370 | . . . . 5 ⊢ ω ∈ V | |
2 | 1 | rabex 3948 | . . . 4 ⊢ {𝑦 ∈ ω ∣ 𝜑} ∈ V |
3 | eleq1 2145 | . . . . 5 ⊢ (𝑥 = {𝑦 ∈ ω ∣ 𝜑} → (𝑥 ∈ Fin ↔ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin)) | |
4 | breq2 3815 | . . . . 5 ⊢ (𝑥 = {𝑦 ∈ ω ∣ 𝜑} → (ω ≼ 𝑥 ↔ ω ≼ {𝑦 ∈ ω ∣ 𝜑})) | |
5 | 3, 4 | orbi12d 740 | . . . 4 ⊢ (𝑥 = {𝑦 ∈ ω ∣ 𝜑} → ((𝑥 ∈ Fin ∨ ω ≼ 𝑥) ↔ ({𝑦 ∈ ω ∣ 𝜑} ∈ Fin ∨ ω ≼ {𝑦 ∈ ω ∣ 𝜑}))) |
6 | inffiexmid.1 | . . . 4 ⊢ (𝑥 ∈ Fin ∨ ω ≼ 𝑥) | |
7 | 2, 5, 6 | vtocl 2664 | . . 3 ⊢ ({𝑦 ∈ ω ∣ 𝜑} ∈ Fin ∨ ω ≼ {𝑦 ∈ ω ∣ 𝜑}) |
8 | ominf 6540 | . . . . . 6 ⊢ ¬ ω ∈ Fin | |
9 | peano1 4371 | . . . . . . . . . 10 ⊢ ∅ ∈ ω | |
10 | elex2 2626 | . . . . . . . . . 10 ⊢ (∅ ∈ ω → ∃𝑤 𝑤 ∈ ω) | |
11 | 9, 10 | ax-mp 7 | . . . . . . . . 9 ⊢ ∃𝑤 𝑤 ∈ ω |
12 | r19.3rmv 3353 | . . . . . . . . 9 ⊢ (∃𝑤 𝑤 ∈ ω → (𝜑 ↔ ∀𝑦 ∈ ω 𝜑)) | |
13 | 11, 12 | ax-mp 7 | . . . . . . . 8 ⊢ (𝜑 ↔ ∀𝑦 ∈ ω 𝜑) |
14 | rabid2 2536 | . . . . . . . 8 ⊢ (ω = {𝑦 ∈ ω ∣ 𝜑} ↔ ∀𝑦 ∈ ω 𝜑) | |
15 | 13, 14 | sylbb2 136 | . . . . . . 7 ⊢ (𝜑 → ω = {𝑦 ∈ ω ∣ 𝜑}) |
16 | 15 | eleq1d 2151 | . . . . . 6 ⊢ (𝜑 → (ω ∈ Fin ↔ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin)) |
17 | 8, 16 | mtbii 632 | . . . . 5 ⊢ (𝜑 → ¬ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin) |
18 | 17 | con2i 590 | . . . 4 ⊢ ({𝑦 ∈ ω ∣ 𝜑} ∈ Fin → ¬ 𝜑) |
19 | infm 6545 | . . . . 5 ⊢ (ω ≼ {𝑦 ∈ ω ∣ 𝜑} → ∃𝑧 𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑}) | |
20 | biidd 170 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜑)) | |
21 | 20 | elrab 2759 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} ↔ (𝑧 ∈ ω ∧ 𝜑)) |
22 | 21 | simprbi 269 | . . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
23 | 22 | exlimiv 1530 | . . . . 5 ⊢ (∃𝑧 𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
24 | 19, 23 | syl 14 | . . . 4 ⊢ (ω ≼ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
25 | 18, 24 | orim12i 709 | . . 3 ⊢ (({𝑦 ∈ ω ∣ 𝜑} ∈ Fin ∨ ω ≼ {𝑦 ∈ ω ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑)) |
26 | 7, 25 | ax-mp 7 | . 2 ⊢ (¬ 𝜑 ∨ 𝜑) |
27 | orcom 680 | . 2 ⊢ ((¬ 𝜑 ∨ 𝜑) ↔ (𝜑 ∨ ¬ 𝜑)) | |
28 | 26, 27 | mpbi 143 | 1 ⊢ (𝜑 ∨ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 103 ∨ wo 662 = wceq 1285 ∃wex 1422 ∈ wcel 1434 ∀wral 2353 {crab 2357 ∅c0 3269 class class class wbr 3811 ωcom 4367 ≼ cdom 6384 Fincfn 6385 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-tr 3902 df-id 4083 df-iord 4156 df-on 4158 df-suc 4161 df-iom 4368 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 df-er 6220 df-en 6386 df-dom 6387 df-fin 6388 |
This theorem is referenced by: (None) |
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