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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 4515 | . . . . 5 ⊢ ω ∈ V | |
2 | 1 | rabex 4080 | . . . 4 ⊢ {𝑦 ∈ ω ∣ 𝜑} ∈ V |
3 | eleq1 2203 | . . . . 5 ⊢ (𝑥 = {𝑦 ∈ ω ∣ 𝜑} → (𝑥 ∈ Fin ↔ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin)) | |
4 | breq2 3941 | . . . . 5 ⊢ (𝑥 = {𝑦 ∈ ω ∣ 𝜑} → (ω ≼ 𝑥 ↔ ω ≼ {𝑦 ∈ ω ∣ 𝜑})) | |
5 | 3, 4 | orbi12d 783 | . . . 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 2743 | . . 3 ⊢ ({𝑦 ∈ ω ∣ 𝜑} ∈ Fin ∨ ω ≼ {𝑦 ∈ ω ∣ 𝜑}) |
8 | ominf 6798 | . . . . . 6 ⊢ ¬ ω ∈ Fin | |
9 | peano1 4516 | . . . . . . . . . 10 ⊢ ∅ ∈ ω | |
10 | elex2 2705 | . . . . . . . . . 10 ⊢ (∅ ∈ ω → ∃𝑤 𝑤 ∈ ω) | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . 9 ⊢ ∃𝑤 𝑤 ∈ ω |
12 | r19.3rmv 3458 | . . . . . . . . 9 ⊢ (∃𝑤 𝑤 ∈ ω → (𝜑 ↔ ∀𝑦 ∈ ω 𝜑)) | |
13 | 11, 12 | ax-mp 5 | . . . . . . . 8 ⊢ (𝜑 ↔ ∀𝑦 ∈ ω 𝜑) |
14 | rabid2 2610 | . . . . . . . 8 ⊢ (ω = {𝑦 ∈ ω ∣ 𝜑} ↔ ∀𝑦 ∈ ω 𝜑) | |
15 | 13, 14 | sylbb2 137 | . . . . . . 7 ⊢ (𝜑 → ω = {𝑦 ∈ ω ∣ 𝜑}) |
16 | 15 | eleq1d 2209 | . . . . . 6 ⊢ (𝜑 → (ω ∈ Fin ↔ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin)) |
17 | 8, 16 | mtbii 664 | . . . . 5 ⊢ (𝜑 → ¬ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin) |
18 | 17 | con2i 617 | . . . 4 ⊢ ({𝑦 ∈ ω ∣ 𝜑} ∈ Fin → ¬ 𝜑) |
19 | infm 6806 | . . . . 5 ⊢ (ω ≼ {𝑦 ∈ ω ∣ 𝜑} → ∃𝑧 𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑}) | |
20 | biidd 171 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜑)) | |
21 | 20 | elrab 2844 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} ↔ (𝑧 ∈ ω ∧ 𝜑)) |
22 | 21 | simprbi 273 | . . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
23 | 22 | exlimiv 1578 | . . . . 5 ⊢ (∃𝑧 𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
24 | 19, 23 | syl 14 | . . . 4 ⊢ (ω ≼ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
25 | 18, 24 | orim12i 749 | . . 3 ⊢ (({𝑦 ∈ ω ∣ 𝜑} ∈ Fin ∨ ω ≼ {𝑦 ∈ ω ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑)) |
26 | 7, 25 | ax-mp 5 | . 2 ⊢ (¬ 𝜑 ∨ 𝜑) |
27 | orcom 718 | . 2 ⊢ ((¬ 𝜑 ∨ 𝜑) ↔ (𝜑 ∨ ¬ 𝜑)) | |
28 | 26, 27 | mpbi 144 | 1 ⊢ (𝜑 ∨ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 104 ∨ wo 698 = wceq 1332 ∃wex 1469 ∈ wcel 1481 ∀wral 2417 {crab 2421 ∅c0 3368 class class class wbr 3937 ωcom 4512 ≼ cdom 6641 Fincfn 6642 |
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 1424 ax-7 1425 ax-gen 1426 ax-ie1 1470 ax-ie2 1471 ax-8 1483 ax-10 1484 ax-11 1485 ax-i12 1486 ax-bndl 1487 ax-4 1488 ax-13 1492 ax-14 1493 ax-17 1507 ax-i9 1511 ax-ial 1515 ax-i5r 1516 ax-ext 2122 ax-sep 4054 ax-nul 4062 ax-pow 4106 ax-pr 4139 ax-un 4363 ax-setind 4460 ax-iinf 4510 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1335 df-fal 1338 df-nf 1438 df-sb 1737 df-eu 2003 df-mo 2004 df-clab 2127 df-cleq 2133 df-clel 2136 df-nfc 2271 df-ne 2310 df-ral 2422 df-rex 2423 df-rab 2426 df-v 2691 df-sbc 2914 df-dif 3078 df-un 3080 df-in 3082 df-ss 3089 df-nul 3369 df-pw 3517 df-sn 3538 df-pr 3539 df-op 3541 df-uni 3745 df-int 3780 df-br 3938 df-opab 3998 df-tr 4035 df-id 4223 df-iord 4296 df-on 4298 df-suc 4301 df-iom 4513 df-xp 4553 df-rel 4554 df-cnv 4555 df-co 4556 df-dm 4557 df-rn 4558 df-res 4559 df-ima 4560 df-iota 5096 df-fun 5133 df-fn 5134 df-f 5135 df-f1 5136 df-fo 5137 df-f1o 5138 df-fv 5139 df-er 6437 df-en 6643 df-dom 6644 df-fin 6645 |
This theorem is referenced by: (None) |
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