Theorem ssfilemd 7064

Description: Lemma for ssfiexmidt 7065. (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 6934	. . 3 ⊢ ({𝑧 ∈ {∅} ∣ 𝜓} ∈ Fin ↔ ∃𝑛 ∈ ω {𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛)
3	1, 2	sylib 122	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ω {𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛)
4		0elnn 4717	. . . . 5 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
5		breq2 4092	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ↔ {𝑧 ∈ {∅} ∣ 𝜓} ≈ ∅))
6		en0 6969	. . . . . . . . . 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 3524	. . . . . . . . 9 ⊢ ({𝑧 ∈ {∅} ∣ 𝜓} = ∅ ↔ ∀𝑧 ∈ {∅} ¬ 𝜓)
10		0ex 4216	. . . . . . . . . . 11 ⊢ ∅ ∈ V
11	10	snm 3792	. . . . . . . . . 10 ⊢ ∃𝑤 𝑤 ∈ {∅}
12		r19.3rmv 3585	. . . . . . . . . 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 741	. . . . . 6 ⊢ (({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ∧ 𝑛 = ∅) → (𝜓 ∨ ¬ 𝜓))
17		ensym 6955	. . . . . . . 8 ⊢ ({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 → 𝑛 ≈ {𝑧 ∈ {∅} ∣ 𝜓})
18		elex2 2819	. . . . . . . 8 ⊢ (∅ ∈ 𝑛 → ∃𝑥 𝑥 ∈ 𝑛)
19		enm 7004	. . . . . . . 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 2962	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓} ↔ (𝑦 ∈ {∅} ∧ 𝜓))
23	22	simprbi 275	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓} → 𝜓)
24	23	orcd 740	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓} → (𝜓 ∨ ¬ 𝜓))
25	24	exlimiv 1646	. . . . . . 7 ⊢ (∃𝑦 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓} → (𝜓 ∨ ¬ 𝜓))
26	20, 25	syl 14	. . . . . 6 ⊢ (({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ∧ ∅ ∈ 𝑛) → (𝜓 ∨ ¬ 𝜓))
27	16, 26	jaodan 804	. . . . 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 2645	. 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 715   = wceq 1397  ∃wex 1540   ∈ wcel 2202  ∀wral 2510  ∃wrex 2511  {crab 2514  ∅c0 3494  {csn 3669   class class class wbr 4088  ωcom 4688   ≈ cen 6907  Fincfn 6909

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-id 4390  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-er 6702  df-en 6910  df-fin 6912

This theorem is referenced by:  ssfiexmidt  7065