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Mirrors > Home > ILE Home > Th. List > ssfiexmid | GIF version |
Description: If any subset of a finite set is finite, excluded middle follows. One direction of Theorem 2.1 of [Bauer], p. 485. (Contributed by Jim Kingdon, 19-May-2020.) |
Ref | Expression |
---|---|
ssfiexmid.1 | ⊢ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) |
Ref | Expression |
---|---|
ssfiexmid | ⊢ (𝜑 ∨ ¬ 𝜑) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 4055 | . . . 4 ⊢ ∅ ∈ V | |
2 | snfig 6708 | . . . 4 ⊢ (∅ ∈ V → {∅} ∈ Fin) | |
3 | 1, 2 | ax-mp 5 | . . 3 ⊢ {∅} ∈ Fin |
4 | ssrab2 3182 | . . 3 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} | |
5 | ssfiexmid.1 | . . . . 5 ⊢ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) | |
6 | p0ex 4112 | . . . . . 6 ⊢ {∅} ∈ V | |
7 | eleq1 2202 | . . . . . . . . 9 ⊢ (𝑥 = {∅} → (𝑥 ∈ Fin ↔ {∅} ∈ Fin)) | |
8 | sseq2 3121 | . . . . . . . . 9 ⊢ (𝑥 = {∅} → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ {∅})) | |
9 | 7, 8 | anbi12d 464 | . . . . . . . 8 ⊢ (𝑥 = {∅} → ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) ↔ ({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}))) |
10 | 9 | imbi1d 230 | . . . . . . 7 ⊢ (𝑥 = {∅} → (((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) ↔ (({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin))) |
11 | 10 | albidv 1796 | . . . . . 6 ⊢ (𝑥 = {∅} → (∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) ↔ ∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin))) |
12 | 6, 11 | spcv 2779 | . . . . 5 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) → ∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin)) |
13 | 5, 12 | ax-mp 5 | . . . 4 ⊢ ∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin) |
14 | 6 | rabex 4072 | . . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ V |
15 | sseq1 3120 | . . . . . . 7 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅})) | |
16 | 15 | anbi2d 459 | . . . . . 6 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) ↔ ({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))) |
17 | eleq1 2202 | . . . . . 6 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ∈ Fin ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin)) | |
18 | 16, 17 | imbi12d 233 | . . . . 5 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → ((({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin) ↔ (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin))) |
19 | 14, 18 | spcv 2779 | . . . 4 ⊢ (∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin) → (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin)) |
20 | 13, 19 | ax-mp 5 | . . 3 ⊢ (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin) |
21 | 3, 4, 20 | mp2an 422 | . 2 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin |
22 | 21 | ssfilem 6769 | 1 ⊢ (𝜑 ∨ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 697 ∀wal 1329 = wceq 1331 ∈ wcel 1480 {crab 2420 Vcvv 2686 ⊆ wss 3071 ∅c0 3363 {csn 3527 Fincfn 6634 |
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 2121 ax-sep 4046 ax-nul 4054 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-iinf 4502 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-suc 4293 df-iom 4505 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-f1 5128 df-fo 5129 df-f1o 5130 df-fv 5131 df-1o 6313 df-er 6429 df-en 6635 df-fin 6637 |
This theorem is referenced by: infiexmid 6771 |
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