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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 4502 | . . . . 5 ⊢ ω ∈ V | |
2 | 1 | rabex 4067 | . . . 4 ⊢ {𝑦 ∈ ω ∣ 𝜑} ∈ V |
3 | eleq1 2200 | . . . . 5 ⊢ (𝑥 = {𝑦 ∈ ω ∣ 𝜑} → (𝑥 ∈ Fin ↔ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin)) | |
4 | breq2 3928 | . . . . 5 ⊢ (𝑥 = {𝑦 ∈ ω ∣ 𝜑} → (ω ≼ 𝑥 ↔ ω ≼ {𝑦 ∈ ω ∣ 𝜑})) | |
5 | 3, 4 | orbi12d 782 | . . . 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 2735 | . . 3 ⊢ ({𝑦 ∈ ω ∣ 𝜑} ∈ Fin ∨ ω ≼ {𝑦 ∈ ω ∣ 𝜑}) |
8 | ominf 6783 | . . . . . 6 ⊢ ¬ ω ∈ Fin | |
9 | peano1 4503 | . . . . . . . . . 10 ⊢ ∅ ∈ ω | |
10 | elex2 2697 | . . . . . . . . . 10 ⊢ (∅ ∈ ω → ∃𝑤 𝑤 ∈ ω) | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . 9 ⊢ ∃𝑤 𝑤 ∈ ω |
12 | r19.3rmv 3448 | . . . . . . . . 9 ⊢ (∃𝑤 𝑤 ∈ ω → (𝜑 ↔ ∀𝑦 ∈ ω 𝜑)) | |
13 | 11, 12 | ax-mp 5 | . . . . . . . 8 ⊢ (𝜑 ↔ ∀𝑦 ∈ ω 𝜑) |
14 | rabid2 2605 | . . . . . . . 8 ⊢ (ω = {𝑦 ∈ ω ∣ 𝜑} ↔ ∀𝑦 ∈ ω 𝜑) | |
15 | 13, 14 | sylbb2 137 | . . . . . . 7 ⊢ (𝜑 → ω = {𝑦 ∈ ω ∣ 𝜑}) |
16 | 15 | eleq1d 2206 | . . . . . 6 ⊢ (𝜑 → (ω ∈ Fin ↔ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin)) |
17 | 8, 16 | mtbii 663 | . . . . 5 ⊢ (𝜑 → ¬ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin) |
18 | 17 | con2i 616 | . . . 4 ⊢ ({𝑦 ∈ ω ∣ 𝜑} ∈ Fin → ¬ 𝜑) |
19 | infm 6791 | . . . . 5 ⊢ (ω ≼ {𝑦 ∈ ω ∣ 𝜑} → ∃𝑧 𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑}) | |
20 | biidd 171 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜑)) | |
21 | 20 | elrab 2835 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} ↔ (𝑧 ∈ ω ∧ 𝜑)) |
22 | 21 | simprbi 273 | . . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
23 | 22 | exlimiv 1577 | . . . . 5 ⊢ (∃𝑧 𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
24 | 19, 23 | syl 14 | . . . 4 ⊢ (ω ≼ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
25 | 18, 24 | orim12i 748 | . . 3 ⊢ (({𝑦 ∈ ω ∣ 𝜑} ∈ Fin ∨ ω ≼ {𝑦 ∈ ω ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑)) |
26 | 7, 25 | ax-mp 5 | . 2 ⊢ (¬ 𝜑 ∨ 𝜑) |
27 | orcom 717 | . 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 697 = wceq 1331 ∃wex 1468 ∈ wcel 1480 ∀wral 2414 {crab 2418 ∅c0 3358 class class class wbr 3924 ωcom 4499 ≼ cdom 6626 Fincfn 6627 |
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 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-nul 4049 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-iinf 4497 |
This theorem depends on definitions: df-bi 116 df-dc 820 df-3or 963 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-tr 4022 df-id 4210 df-iord 4283 df-on 4285 df-suc 4288 df-iom 4500 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-f1 5123 df-fo 5124 df-f1o 5125 df-fv 5126 df-er 6422 df-en 6628 df-dom 6629 df-fin 6630 |
This theorem is referenced by: (None) |
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