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Theorem ssfiexmid 6336
 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.)
Hypothesis
Ref Expression
ssfiexmid.1 𝑥𝑦((𝑥 ∈ Fin ∧ 𝑦𝑥) → 𝑦 ∈ Fin)
Assertion
Ref Expression
ssfiexmid (𝜑 ∨ ¬ 𝜑)
Distinct variable group:   𝜑,𝑥,𝑦

Proof of Theorem ssfiexmid
Dummy variables 𝑛 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 0ex 3884 . . . . 5 ∅ ∈ V
2 snfig 6291 . . . . 5 (∅ ∈ V → {∅} ∈ Fin)
31, 2ax-mp 7 . . . 4 {∅} ∈ Fin
4 ssrab2 3025 . . . 4 {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}
5 ssfiexmid.1 . . . . . 6 𝑥𝑦((𝑥 ∈ Fin ∧ 𝑦𝑥) → 𝑦 ∈ Fin)
6 p0ex 3939 . . . . . . 7 {∅} ∈ V
7 eleq1 2100 . . . . . . . . . 10 (𝑥 = {∅} → (𝑥 ∈ Fin ↔ {∅} ∈ Fin))
8 sseq2 2967 . . . . . . . . . 10 (𝑥 = {∅} → (𝑦𝑥𝑦 ⊆ {∅}))
97, 8anbi12d 442 . . . . . . . . 9 (𝑥 = {∅} → ((𝑥 ∈ Fin ∧ 𝑦𝑥) ↔ ({∅} ∈ Fin ∧ 𝑦 ⊆ {∅})))
109imbi1d 220 . . . . . . . 8 (𝑥 = {∅} → (((𝑥 ∈ Fin ∧ 𝑦𝑥) → 𝑦 ∈ Fin) ↔ (({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin)))
1110albidv 1705 . . . . . . 7 (𝑥 = {∅} → (∀𝑦((𝑥 ∈ Fin ∧ 𝑦𝑥) → 𝑦 ∈ Fin) ↔ ∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin)))
126, 11spcv 2646 . . . . . 6 (∀𝑥𝑦((𝑥 ∈ Fin ∧ 𝑦𝑥) → 𝑦 ∈ Fin) → ∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin))
135, 12ax-mp 7 . . . . 5 𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin)
146rabex 3901 . . . . . 6 {𝑧 ∈ {∅} ∣ 𝜑} ∈ V
15 sseq1 2966 . . . . . . . 8 (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ⊆ {∅} ↔ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}))
1615anbi2d 437 . . . . . . 7 (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) ↔ ({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅})))
17 eleq1 2100 . . . . . . 7 (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑦 ∈ Fin ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin))
1816, 17imbi12d 223 . . . . . 6 (𝑦 = {𝑧 ∈ {∅} ∣ 𝜑} → ((({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin) ↔ (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin)))
1914, 18spcv 2646 . . . . 5 (∀𝑦(({∅} ∈ Fin ∧ 𝑦 ⊆ {∅}) → 𝑦 ∈ Fin) → (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin))
2013, 19ax-mp 7 . . . 4 (({∅} ∈ Fin ∧ {𝑧 ∈ {∅} ∣ 𝜑} ⊆ {∅}) → {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin)
213, 4, 20mp2an 402 . . 3 {𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin
22 isfi 6241 . . 3 ({𝑧 ∈ {∅} ∣ 𝜑} ∈ Fin ↔ ∃𝑛 ∈ ω {𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛)
2321, 22mpbi 133 . 2 𝑛 ∈ ω {𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛
24 0elnn 4340 . . . . 5 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
25 breq2 3768 . . . . . . . . . 10 (𝑛 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ↔ {𝑧 ∈ {∅} ∣ 𝜑} ≈ ∅))
26 en0 6275 . . . . . . . . . 10 ({𝑧 ∈ {∅} ∣ 𝜑} ≈ ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
2725, 26syl6bb 185 . . . . . . . . 9 (𝑛 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅))
2827biimpac 282 . . . . . . . 8 (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛𝑛 = ∅) → {𝑧 ∈ {∅} ∣ 𝜑} = ∅)
29 rabeq0 3247 . . . . . . . . 9 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ↔ ∀𝑧 ∈ {∅} ¬ 𝜑)
301snm 3488 . . . . . . . . . 10 𝑤 𝑤 ∈ {∅}
31 r19.3rmv 3312 . . . . . . . . . 10 (∃𝑤 𝑤 ∈ {∅} → (¬ 𝜑 ↔ ∀𝑧 ∈ {∅} ¬ 𝜑))
3230, 31ax-mp 7 . . . . . . . . 9 𝜑 ↔ ∀𝑧 ∈ {∅} ¬ 𝜑)
3329, 32bitr4i 176 . . . . . . . 8 ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ↔ ¬ 𝜑)
3428, 33sylib 127 . . . . . . 7 (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛𝑛 = ∅) → ¬ 𝜑)
3534olcd 653 . . . . . 6 (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛𝑛 = ∅) → (𝜑 ∨ ¬ 𝜑))
36 ensym 6261 . . . . . . . 8 ({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛𝑛 ≈ {𝑧 ∈ {∅} ∣ 𝜑})
37 elex2 2570 . . . . . . . 8 (∅ ∈ 𝑛 → ∃𝑥 𝑥𝑛)
38 enm 6294 . . . . . . . 8 ((𝑛 ≈ {𝑧 ∈ {∅} ∣ 𝜑} ∧ ∃𝑥 𝑥𝑛) → ∃𝑦 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑})
3936, 37, 38syl2an 273 . . . . . . 7 (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ∧ ∅ ∈ 𝑛) → ∃𝑦 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑})
40 biidd 161 . . . . . . . . . . 11 (𝑧 = 𝑦 → (𝜑𝜑))
4140elrab 2698 . . . . . . . . . 10 (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ (𝑦 ∈ {∅} ∧ 𝜑))
4241simprbi 260 . . . . . . . . 9 (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
4342orcd 652 . . . . . . . 8 (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑} → (𝜑 ∨ ¬ 𝜑))
4443exlimiv 1489 . . . . . . 7 (∃𝑦 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜑} → (𝜑 ∨ ¬ 𝜑))
4539, 44syl 14 . . . . . 6 (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ∧ ∅ ∈ 𝑛) → (𝜑 ∨ ¬ 𝜑))
4635, 45jaodan 710 . . . . 5 (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 ∧ (𝑛 = ∅ ∨ ∅ ∈ 𝑛)) → (𝜑 ∨ ¬ 𝜑))
4724, 46sylan2 270 . . . 4 (({𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛𝑛 ∈ ω) → (𝜑 ∨ ¬ 𝜑))
4847ancoms 255 . . 3 ((𝑛 ∈ ω ∧ {𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛) → (𝜑 ∨ ¬ 𝜑))
4948rexlimiva 2428 . 2 (∃𝑛 ∈ ω {𝑧 ∈ {∅} ∣ 𝜑} ≈ 𝑛 → (𝜑 ∨ ¬ 𝜑))
5023, 49ax-mp 7 1 (𝜑 ∨ ¬ 𝜑)
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629  ∀wal 1241   = wceq 1243  ∃wex 1381   ∈ wcel 1393  ∀wral 2306  ∃wrex 2307  {crab 2310  Vcvv 2557   ⊆ wss 2917  ∅c0 3224  {csn 3375   class class class wbr 3764  ωcom 4313   ≈ cen 6219  Fincfn 6221 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-iinf 4311 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-sbc 2765  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-int 3616  df-br 3765  df-opab 3819  df-id 4030  df-suc 4108  df-iom 4314  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910  df-1o 6001  df-er 6106  df-en 6222  df-fin 6224 This theorem is referenced by: (None)
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