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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 4436 | . . . . 5 ⊢ ω ∈ V | |
2 | 1 | rabex 4004 | . . . 4 ⊢ {𝑦 ∈ ω ∣ 𝜑} ∈ V |
3 | eleq1 2157 | . . . . 5 ⊢ (𝑥 = {𝑦 ∈ ω ∣ 𝜑} → (𝑥 ∈ Fin ↔ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin)) | |
4 | breq2 3871 | . . . . 5 ⊢ (𝑥 = {𝑦 ∈ ω ∣ 𝜑} → (ω ≼ 𝑥 ↔ ω ≼ {𝑦 ∈ ω ∣ 𝜑})) | |
5 | 3, 4 | orbi12d 745 | . . . 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 2687 | . . 3 ⊢ ({𝑦 ∈ ω ∣ 𝜑} ∈ Fin ∨ ω ≼ {𝑦 ∈ ω ∣ 𝜑}) |
8 | ominf 6692 | . . . . . 6 ⊢ ¬ ω ∈ Fin | |
9 | peano1 4437 | . . . . . . . . . 10 ⊢ ∅ ∈ ω | |
10 | elex2 2649 | . . . . . . . . . 10 ⊢ (∅ ∈ ω → ∃𝑤 𝑤 ∈ ω) | |
11 | 9, 10 | ax-mp 7 | . . . . . . . . 9 ⊢ ∃𝑤 𝑤 ∈ ω |
12 | r19.3rmv 3392 | . . . . . . . . 9 ⊢ (∃𝑤 𝑤 ∈ ω → (𝜑 ↔ ∀𝑦 ∈ ω 𝜑)) | |
13 | 11, 12 | ax-mp 7 | . . . . . . . 8 ⊢ (𝜑 ↔ ∀𝑦 ∈ ω 𝜑) |
14 | rabid2 2557 | . . . . . . . 8 ⊢ (ω = {𝑦 ∈ ω ∣ 𝜑} ↔ ∀𝑦 ∈ ω 𝜑) | |
15 | 13, 14 | sylbb2 137 | . . . . . . 7 ⊢ (𝜑 → ω = {𝑦 ∈ ω ∣ 𝜑}) |
16 | 15 | eleq1d 2163 | . . . . . 6 ⊢ (𝜑 → (ω ∈ Fin ↔ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin)) |
17 | 8, 16 | mtbii 637 | . . . . 5 ⊢ (𝜑 → ¬ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin) |
18 | 17 | con2i 595 | . . . 4 ⊢ ({𝑦 ∈ ω ∣ 𝜑} ∈ Fin → ¬ 𝜑) |
19 | infm 6700 | . . . . 5 ⊢ (ω ≼ {𝑦 ∈ ω ∣ 𝜑} → ∃𝑧 𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑}) | |
20 | biidd 171 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜑)) | |
21 | 20 | elrab 2785 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} ↔ (𝑧 ∈ ω ∧ 𝜑)) |
22 | 21 | simprbi 270 | . . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
23 | 22 | exlimiv 1541 | . . . . 5 ⊢ (∃𝑧 𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
24 | 19, 23 | syl 14 | . . . 4 ⊢ (ω ≼ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
25 | 18, 24 | orim12i 714 | . . 3 ⊢ (({𝑦 ∈ ω ∣ 𝜑} ∈ Fin ∨ ω ≼ {𝑦 ∈ ω ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑)) |
26 | 7, 25 | ax-mp 7 | . 2 ⊢ (¬ 𝜑 ∨ 𝜑) |
27 | orcom 685 | . 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 667 = wceq 1296 ∃wex 1433 ∈ wcel 1445 ∀wral 2370 {crab 2374 ∅c0 3302 class class class wbr 3867 ωcom 4433 ≼ cdom 6536 Fincfn 6537 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 582 ax-in2 583 ax-io 668 ax-5 1388 ax-7 1389 ax-gen 1390 ax-ie1 1434 ax-ie2 1435 ax-8 1447 ax-10 1448 ax-11 1449 ax-i12 1450 ax-bndl 1451 ax-4 1452 ax-13 1456 ax-14 1457 ax-17 1471 ax-i9 1475 ax-ial 1479 ax-i5r 1480 ax-ext 2077 ax-sep 3978 ax-nul 3986 ax-pow 4030 ax-pr 4060 ax-un 4284 ax-setind 4381 ax-iinf 4431 |
This theorem depends on definitions: df-bi 116 df-dc 784 df-3or 928 df-3an 929 df-tru 1299 df-fal 1302 df-nf 1402 df-sb 1700 df-eu 1958 df-mo 1959 df-clab 2082 df-cleq 2088 df-clel 2091 df-nfc 2224 df-ne 2263 df-ral 2375 df-rex 2376 df-rab 2379 df-v 2635 df-sbc 2855 df-dif 3015 df-un 3017 df-in 3019 df-ss 3026 df-nul 3303 df-pw 3451 df-sn 3472 df-pr 3473 df-op 3475 df-uni 3676 df-int 3711 df-br 3868 df-opab 3922 df-tr 3959 df-id 4144 df-iord 4217 df-on 4219 df-suc 4222 df-iom 4434 df-xp 4473 df-rel 4474 df-cnv 4475 df-co 4476 df-dm 4477 df-rn 4478 df-res 4479 df-ima 4480 df-iota 5014 df-fun 5051 df-fn 5052 df-f 5053 df-f1 5054 df-fo 5055 df-f1o 5056 df-fv 5057 df-er 6332 df-en 6538 df-dom 6539 df-fin 6540 |
This theorem is referenced by: (None) |
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