Theorem ssfilemd 7131

Description: Lemma for ssfiexmidt 7132. (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 6999	. . 3 ⊢ ({𝑧 ∈ {∅} ∣ 𝜓} ∈ Fin ↔ ∃𝑛 ∈ ω {𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛)
3	1, 2	sylib 122	. 2 ⊢ (𝜑 → ∃𝑛 ∈ ω {𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛)
4		0elnn 4740	. . . . 5 ⊢ (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∅ ∈ 𝑛))
5		breq2 4112	. . . . . . . . . 10 ⊢ (𝑛 = ∅ → ({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ↔ {𝑧 ∈ {∅} ∣ 𝜓} ≈ ∅))
6		en0 7034	. . . . . . . . . 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 3537	. . . . . . . . 9 ⊢ ({𝑧 ∈ {∅} ∣ 𝜓} = ∅ ↔ ∀𝑧 ∈ {∅} ¬ 𝜓)
10		0ex 4236	. . . . . . . . . . 11 ⊢ ∅ ∈ V
11	10	snm 3811	. . . . . . . . . 10 ⊢ ∃𝑤 𝑤 ∈ {∅}
12		r19.3rmv 3599	. . . . . . . . . 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 742	. . . . . 6 ⊢ (({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ∧ 𝑛 = ∅) → (𝜓 ∨ ¬ 𝜓))
17		ensym 7020	. . . . . . . 8 ⊢ ({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 → 𝑛 ≈ {𝑧 ∈ {∅} ∣ 𝜓})
18		elex2 2829	. . . . . . . 8 ⊢ (∅ ∈ 𝑛 → ∃𝑥 𝑥 ∈ 𝑛)
19		enm 7070	. . . . . . . 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 2972	. . . . . . . . . 10 ⊢ (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓} ↔ (𝑦 ∈ {∅} ∧ 𝜓))
23	22	simprbi 275	. . . . . . . . 9 ⊢ (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓} → 𝜓)
24	23	orcd 741	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓} → (𝜓 ∨ ¬ 𝜓))
25	24	exlimiv 1647	. . . . . . 7 ⊢ (∃𝑦 𝑦 ∈ {𝑧 ∈ {∅} ∣ 𝜓} → (𝜓 ∨ ¬ 𝜓))
26	20, 25	syl 14	. . . . . 6 ⊢ (({𝑧 ∈ {∅} ∣ 𝜓} ≈ 𝑛 ∧ ∅ ∈ 𝑛) → (𝜓 ∨ ¬ 𝜓))
27	16, 26	jaodan 805	. . . . 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 2655	. 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 716   = wceq 1398  ∃wex 1541   ∈ wcel 2203  ∀wral 2520  ∃wrex 2521  {crab 2524  ∅c0 3507  {csn 3688   class class class wbr 4108  ωcom 4711   ≈ cen 6972  Fincfn 6974

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2205  ax-14 2206  ax-ext 2214  ax-sep 4227  ax-nul 4235  ax-pow 4286  ax-pr 4321  ax-un 4553  ax-iinf 4709

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2083  df-mo 2084  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-ral 2525  df-rex 2526  df-rab 2529  df-v 2814  df-sbc 3042  df-dif 3212  df-un 3214  df-in 3216  df-ss 3223  df-nul 3508  df-pw 3670  df-sn 3694  df-pr 3695  df-op 3697  df-uni 3914  df-int 3949  df-br 4109  df-opab 4171  df-id 4413  df-suc 4491  df-iom 4712  df-xp 4754  df-rel 4755  df-cnv 4756  df-co 4757  df-dm 4758  df-rn 4759  df-res 4760  df-ima 4761  df-iota 5311  df-fun 5353  df-fn 5354  df-f 5355  df-f1 5356  df-fo 5357  df-f1o 5358  df-fv 5359  df-er 6766  df-en 6975  df-fin 6977

This theorem is referenced by:  ssfiexmidt  7132