Theorem ssfilemd 7173

Description: Lemma for ssfiexmidt 7174. (Contributed by Jim Kingdon, 3-Feb-2022.)

Hypothesis

Ref	Expression
ssfilemd.1	⊢ (𝜑 → {𝑧 ∈ {∅} ∣ 𝜓} ∈ Fin)

Assertion

Ref	Expression
ssfilemd	⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓))

Distinct variable group:   𝜓,𝑧

Allowed substitution hint:   𝜑(𝑧)

Proof of Theorem ssfilemd

Dummy variables 𝑛 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssfilemd.1	. . 3 ⊢ (𝜑 → {𝑧 ∈ {∅} ∣ 𝜓} ∈ Fin)
2		isfi 7041	. . 3 ⊢ ({𝑧 ∈ {∅} ∣ 𝜓} ∈ Fin ↔ ∃𝑛 ∈ ω {𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛)
3	1, 2	sylib 122	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ω {𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛)
4		0elnn 4764	. . . . 5 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
5		breq2 4132	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ↔ {𝑧 ∈ {∅} ∣ 𝜓} ≈ ∅))
6		en0 7076	. . . . . . . . . 10 ⊢ ({𝑧 ∈ {∅} ∣ 𝜓} ≈ ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜓} = ∅)
7	5, 6	bitrdi 196	. . . . . . . . 9 ⊢ (𝑛 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ↔ {𝑧 ∈ {∅} ∣ 𝜓} = ∅))
8	7	biimpac 298	. . . . . . . 8 ⊢ (({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ∧ 𝑛 = ∅) → {𝑧 ∈ {∅} ∣ 𝜓} = ∅)
9		rabeq0 3552	. . . . . . . . 9 ⊢ ({𝑧 ∈ {∅} ∣ 𝜓} = ∅ ↔ ∀𝑧 ∈ {∅} ¬ 𝜓)
10		0ex 4258	. . . . . . . . . . 11 ⊢ ∅ ∈ V
11	10	snm 3831	. . . . . . . . . 10 ⊢ ∃𝑤 𝑤 ∈ {∅}
12		r19.3rmv 3618	. . . . . . . . . 10 ⊢ (∃𝑤 𝑤 ∈ {∅} → (¬ 𝜓 ↔ ∀𝑧 ∈ {∅} ¬ 𝜓))
13	11, 12	ax-mp 5	. . . . . . . . 9 ⊢ (¬ 𝜓 ↔ ∀𝑧 ∈ {∅} ¬ 𝜓)
14	9, 13	bitr4i 187	. . . . . . . 8 ⊢ ({𝑧 ∈ {∅} ∣ 𝜓} = ∅ ↔ ¬ 𝜓)
15	8, 14	sylib 122	. . . . . . 7 ⊢ (({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ∧ 𝑛 = ∅) → ¬ 𝜓)
16	15	olcd 746	. . . . . 6 ⊢ (({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ∧ 𝑛 = ∅) → (𝜓 ∨ ¬ 𝜓))
17		ensym 7062	. . . . . . . 8 ⊢ ({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 → 𝑛 ≈ {𝑧 ∈ {∅} ∣ 𝜓})
18		elex2 2838	. . . . . . . 8 ⊢ (∅ ∈ 𝑛 → ∃𝑥 𝑥 ∈ 𝑛)
19		enm 7112	. . . . . . . 8 ⊢ ((𝑛 ≈ {𝑧 ∈ {∅} ∣ 𝜓} ∧ ∃𝑥 𝑥 ∈ 𝑛) → ∃𝑦 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓})
20	17, 18, 19	syl2an 289	. . . . . . 7 ⊢ (({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ∧ ∅ ∈ 𝑛) → ∃𝑦 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓})
21		biidd 172	. . . . . . . . . . 11 ⊢ (𝑧 = 𝑦 → (𝜓 ↔ 𝜓))
22	21	elrab 2982	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓} ↔ (𝑦 ∈ {∅} ∧ 𝜓))
23	22	simprbi 275	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓} → 𝜓)
24	23	orcd 745	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓} → (𝜓 ∨ ¬ 𝜓))
25	24	exlimiv 1651	. . . . . . 7 ⊢ (∃𝑦 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓} → (𝜓 ∨ ¬ 𝜓))
26	20, 25	syl 14	. . . . . 6 ⊢ (({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ∧ ∅ ∈ 𝑛) → (𝜓 ∨ ¬ 𝜓))
27	16, 26	jaodan 809	. . . . 5 ⊢ (({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ∧ (𝑛 = ∅ ∨ ∅ ∈ 𝑛)) → (𝜓 ∨ ¬ 𝜓))
28	4, 27	sylan2 286	. . . 4 ⊢ (({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ∧ 𝑛 ∈ ω) → (𝜓 ∨ ¬ 𝜓))
29	28	ancoms 268	. . 3 ⊢ ((𝑛 ∈ ω ∧ {𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛) → (𝜓 ∨ ¬ 𝜓))
30	29	rexlimiva 2663	. 2 ⊢ (∃𝑛 ∈ ω {𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 → (𝜓 ∨ ¬ 𝜓))
31	3, 30	syl 14	1 ⊢ (𝜑 → (𝜓 ∨ ¬ 𝜓))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 720   = wceq 1402  ∃wex 1545   ∈ wcel 2209  ∀wral 2528  ∃wrex 2529  {crab 2532  ∅c0 3520  {csn 3708   class class class wbr 4128  ωcom 4735   ≈ cen 7014  Fincfn 7016

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4247  ax-nul 4257  ax-pow 4309  ax-pr 4344  ax-un 4576  ax-iinf 4733

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-nul 3521  df-pw 3690  df-sn 3714  df-pr 3715  df-op 3717  df-uni 3934  df-int 3969  df-br 4129  df-opab 4191  df-id 4436  df-suc 4514  df-iom 4736  df-xp 4778  df-rel 4779  df-cnv 4780  df-co 4781  df-dm 4782  df-rn 4783  df-res 4784  df-ima 4785  df-iota 5335  df-fun 5377  df-fn 5378  df-f 5379  df-f1 5380  df-fo 5381  df-f1o 5382  df-fv 5383  df-er 6801  df-en 7017  df-fin 7019

This theorem is referenced by:  ssfiexmidt  7174