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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 3905 | . . . 4 ⊢ ∅ ∈ V | |
2 | snfig 6314 | . . . 4 ⊢ (∅ ∈ V → {∅} ∈ Fin) | |
3 | 1, 2 | ax-mp 7 | . . 3 ⊢ {∅} ∈ Fin |
4 | ssrab2 3079 | . . 3 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅} | |
5 | ssfiexmid.1 | . . . . 5 ⊢ ∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) | |
6 | p0ex 3959 | . . . . . 6 ⊢ {∅} ∈ V | |
7 | eleq1 2141 | . . . . . . . . 9 ⊢ (𝑥 = {∅} → (𝑥 ∈ Fin ↔ {∅} ∈ Fin)) | |
8 | sseq2 3021 | . . . . . . . . 9 ⊢ (𝑥 = {∅} → (𝑦 ⊆ 𝑥 ↔ 𝑦 ⊆ {∅})) | |
9 | 7, 8 | anbi12d 456 | . . . . . . . 8 ⊢ (𝑥 = {∅} → ((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) ↔ ({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}))) |
10 | 9 | imbi1d 229 | . . . . . . 7 ⊢ (𝑥 = {∅} → (((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) ↔ (({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin))) |
11 | 10 | albidv 1745 | . . . . . 6 ⊢ (𝑥 = {∅} → (∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) ↔ ∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin))) |
12 | 6, 11 | spcv 2691 | . . . . 5 ⊢ (∀𝑥∀𝑦((𝑥 ∈ Fin ∧ 𝑦 ⊆ 𝑥) → 𝑦 ∈ Fin) → ∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin)) |
13 | 5, 12 | ax-mp 7 | . . . 4 ⊢ ∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin) |
14 | 6 | rabex 3922 | . . . . 5 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ V |
15 | sseq1 3020 | . . . . . . 7 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅})) | |
16 | 15 | anbi2d 451 | . . . . . 6 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) ↔ ({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))) |
17 | eleq1 2141 | . . . . . 6 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ∈ Fin ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin)) | |
18 | 16, 17 | imbi12d 232 | . . . . 5 ⊢ (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → ((({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin) ↔ (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin))) |
19 | 14, 18 | spcv 2691 | . . . 4 ⊢ (∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin) → (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin)) |
20 | 13, 19 | ax-mp 7 | . . 3 ⊢ (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin) |
21 | 3, 4, 20 | mp2an 416 | . 2 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin |
22 | 21 | ssfilem 6360 | 1 ⊢ (𝜑 ∨ ¬ 𝜑) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 102 ∨ wo 661 ∀wal 1282 = wceq 1284 ∈ wcel 1433 {crab 2352 Vcvv 2601 ⊆ wss 2973 ∅c0 3251 {csn 3398 Fincfn 6244 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 576 ax-in2 577 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-nul 3904 ax-pow 3948 ax-pr 3964 ax-un 4188 ax-iinf 4329 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-fal 1290 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-rab 2357 df-v 2603 df-sbc 2816 df-dif 2975 df-un 2977 df-in 2979 df-ss 2986 df-nul 3252 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-int 3637 df-br 3786 df-opab 3840 df-id 4048 df-suc 4126 df-iom 4332 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-ima 4376 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-f1 4927 df-fo 4928 df-f1o 4929 df-fv 4930 df-1o 6024 df-er 6129 df-en 6245 df-fin 6247 |
This theorem is referenced by: infiexmid 6362 |
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