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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 4591 | . . . . 5 ⊢ ω ∈ V | |
2 | 1 | rabex 4146 | . . . 4 ⊢ {𝑦 ∈ ω ∣ 𝜑} ∈ V |
3 | eleq1 2240 | . . . . 5 ⊢ (𝑥 = {𝑦 ∈ ω ∣ 𝜑} → (𝑥 ∈ Fin ↔ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin)) | |
4 | breq2 4006 | . . . . 5 ⊢ (𝑥 = {𝑦 ∈ ω ∣ 𝜑} → (ω ≼ 𝑥 ↔ ω ≼ {𝑦 ∈ ω ∣ 𝜑})) | |
5 | 3, 4 | orbi12d 793 | . . . 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 2791 | . . 3 ⊢ ({𝑦 ∈ ω ∣ 𝜑} ∈ Fin ∨ ω ≼ {𝑦 ∈ ω ∣ 𝜑}) |
8 | ominf 6893 | . . . . . 6 ⊢ ¬ ω ∈ Fin | |
9 | peano1 4592 | . . . . . . . . . 10 ⊢ ∅ ∈ ω | |
10 | elex2 2753 | . . . . . . . . . 10 ⊢ (∅ ∈ ω → ∃𝑤 𝑤 ∈ ω) | |
11 | 9, 10 | ax-mp 5 | . . . . . . . . 9 ⊢ ∃𝑤 𝑤 ∈ ω |
12 | r19.3rmv 3513 | . . . . . . . . 9 ⊢ (∃𝑤 𝑤 ∈ ω → (𝜑 ↔ ∀𝑦 ∈ ω 𝜑)) | |
13 | 11, 12 | ax-mp 5 | . . . . . . . 8 ⊢ (𝜑 ↔ ∀𝑦 ∈ ω 𝜑) |
14 | rabid2 2653 | . . . . . . . 8 ⊢ (ω = {𝑦 ∈ ω ∣ 𝜑} ↔ ∀𝑦 ∈ ω 𝜑) | |
15 | 13, 14 | sylbb2 138 | . . . . . . 7 ⊢ (𝜑 → ω = {𝑦 ∈ ω ∣ 𝜑}) |
16 | 15 | eleq1d 2246 | . . . . . 6 ⊢ (𝜑 → (ω ∈ Fin ↔ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin)) |
17 | 8, 16 | mtbii 674 | . . . . 5 ⊢ (𝜑 → ¬ {𝑦 ∈ ω ∣ 𝜑} ∈ Fin) |
18 | 17 | con2i 627 | . . . 4 ⊢ ({𝑦 ∈ ω ∣ 𝜑} ∈ Fin → ¬ 𝜑) |
19 | infm 6901 | . . . . 5 ⊢ (ω ≼ {𝑦 ∈ ω ∣ 𝜑} → ∃𝑧 𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑}) | |
20 | biidd 172 | . . . . . . . 8 ⊢ (𝑦 = 𝑧 → (𝜑 ↔ 𝜑)) | |
21 | 20 | elrab 2893 | . . . . . . 7 ⊢ (𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} ↔ (𝑧 ∈ ω ∧ 𝜑)) |
22 | 21 | simprbi 275 | . . . . . 6 ⊢ (𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
23 | 22 | exlimiv 1598 | . . . . 5 ⊢ (∃𝑧 𝑧 ∈ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
24 | 19, 23 | syl 14 | . . . 4 ⊢ (ω ≼ {𝑦 ∈ ω ∣ 𝜑} → 𝜑) |
25 | 18, 24 | orim12i 759 | . . 3 ⊢ (({𝑦 ∈ ω ∣ 𝜑} ∈ Fin ∨ ω ≼ {𝑦 ∈ ω ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑)) |
26 | 7, 25 | ax-mp 5 | . 2 ⊢ (¬ 𝜑 ∨ 𝜑) |
27 | orcom 728 | . 2 ⊢ ((¬ 𝜑 ∨ 𝜑) ↔ (𝜑 ∨ ¬ 𝜑)) | |
28 | 26, 27 | mpbi 145 | 1 ⊢ (𝜑 ∨ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 ↔ wb 105 ∨ wo 708 = wceq 1353 ∃wex 1492 ∈ wcel 2148 ∀wral 2455 {crab 2459 ∅c0 3422 class class class wbr 4002 ωcom 4588 ≼ cdom 6736 Fincfn 6737 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4120 ax-nul 4128 ax-pow 4173 ax-pr 4208 ax-un 4432 ax-setind 4535 ax-iinf 4586 |
This theorem depends on definitions: df-bi 117 df-dc 835 df-3or 979 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3577 df-sn 3598 df-pr 3599 df-op 3601 df-uni 3810 df-int 3845 df-br 4003 df-opab 4064 df-tr 4101 df-id 4292 df-iord 4365 df-on 4367 df-suc 4370 df-iom 4589 df-xp 4631 df-rel 4632 df-cnv 4633 df-co 4634 df-dm 4635 df-rn 4636 df-res 4637 df-ima 4638 df-iota 5177 df-fun 5217 df-fn 5218 df-f 5219 df-f1 5220 df-fo 5221 df-f1o 5222 df-fv 5223 df-er 6532 df-en 6738 df-dom 6739 df-fin 6740 |
This theorem is referenced by: (None) |
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