Theorem cantnflem2 8625

Description: Lemma for cantnf 8628. (Contributed by Mario Carneiro, 28-May-2015.)

Hypotheses

Ref	Expression
cantnfs.s	⊢ 𝑆 = dom (𝐴 CNF 𝐵)
cantnfs.a	⊢ (𝜑 → 𝐴 ∈ On)
cantnfs.b	⊢ (𝜑 → 𝐵 ∈ On)
oemapval.t	⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))}
cantnf.c	⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑𝑜 𝐵))
cantnf.s	⊢ (𝜑 → 𝐶 ⊆ ran (𝐴 CNF 𝐵))
cantnf.e	⊢ (𝜑 → ∅ ∈ 𝐶)

Assertion

Ref	Expression
cantnflem2	⊢ (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)))

Distinct variable groups:   𝑥,𝑤,𝑦,𝑧,𝐵   𝑤,𝐶,𝑥,𝑦,𝑧   𝑤,𝐴,𝑥,𝑦,𝑧   𝑥,𝑆,𝑦,𝑧   𝜑,𝑥,𝑦,𝑧

Allowed substitution hints:   𝜑(𝑤)   𝑆(𝑤)   𝑇(𝑥,𝑦,𝑧,𝑤)

Proof of Theorem cantnflem2

Step	Hyp	Ref	Expression
1		cantnfs.a	. . 3 ⊢ (𝜑 → 𝐴 ∈ On)
2		cantnfs.b	. . . . . . . . . 10 ⊢ (𝜑 → 𝐵 ∈ On)
3		oecl 7662	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)
4	1, 2, 3	syl2anc 694	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑𝑜 𝐵) ∈ On)
5		cantnf.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑𝑜 𝐵))
6		onelon 5786	. . . . . . . . 9 ⊢ (((𝐴 ↑𝑜 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 ↑𝑜 𝐵)) → 𝐶 ∈ On)
7	4, 5, 6	syl2anc 694	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On)
8		cantnf.e	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐶)
9		ondif1 7626	. . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 1𝑜) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
10	7, 8, 9	sylanbrc 699	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (On ∖ 1𝑜))
11	10	eldifbd 3620	. . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ∈ 1𝑜)
12		ssel 3630	. . . . . . 7 ⊢ ((𝐴 ↑𝑜 𝐵) ⊆ 1𝑜 → (𝐶 ∈ (𝐴 ↑𝑜 𝐵) → 𝐶 ∈ 1𝑜))
13	5, 12	syl5com 31	. . . . . 6 ⊢ (𝜑 → ((𝐴 ↑𝑜 𝐵) ⊆ 1𝑜 → 𝐶 ∈ 1𝑜))
14	11, 13	mtod 189	. . . . 5 ⊢ (𝜑 → ¬ (𝐴 ↑𝑜 𝐵) ⊆ 1𝑜)
15		oe0m 7643	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ↑𝑜 𝐵) = (1𝑜 ∖ 𝐵))
16	2, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → (∅ ↑𝑜 𝐵) = (1𝑜 ∖ 𝐵))
17		difss 3770	. . . . . . . 8 ⊢ (1𝑜 ∖ 𝐵) ⊆ 1𝑜
18	16, 17	syl6eqss 3688	. . . . . . 7 ⊢ (𝜑 → (∅ ↑𝑜 𝐵) ⊆ 1𝑜)
19		oveq1 6697	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 𝐵))
20	19	sseq1d 3665	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐴 ↑𝑜 𝐵) ⊆ 1𝑜 ↔ (∅ ↑𝑜 𝐵) ⊆ 1𝑜))
21	18, 20	syl5ibrcom 237	. . . . . 6 ⊢ (𝜑 → (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) ⊆ 1𝑜))
22		oe1m 7670	. . . . . . . 8 ⊢ (𝐵 ∈ On → (1𝑜 ↑𝑜 𝐵) = 1𝑜)
23		eqimss 3690	. . . . . . . 8 ⊢ ((1𝑜 ↑𝑜 𝐵) = 1𝑜 → (1𝑜 ↑𝑜 𝐵) ⊆ 1𝑜)
24	2, 22, 23	3syl 18	. . . . . . 7 ⊢ (𝜑 → (1𝑜 ↑𝑜 𝐵) ⊆ 1𝑜)
25		oveq1 6697	. . . . . . . 8 ⊢ (𝐴 = 1𝑜 → (𝐴 ↑𝑜 𝐵) = (1𝑜 ↑𝑜 𝐵))
26	25	sseq1d 3665	. . . . . . 7 ⊢ (𝐴 = 1𝑜 → ((𝐴 ↑𝑜 𝐵) ⊆ 1𝑜 ↔ (1𝑜 ↑𝑜 𝐵) ⊆ 1𝑜))
27	24, 26	syl5ibrcom 237	. . . . . 6 ⊢ (𝜑 → (𝐴 = 1𝑜 → (𝐴 ↑𝑜 𝐵) ⊆ 1𝑜))
28	21, 27	jaod 394	. . . . 5 ⊢ (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → (𝐴 ↑𝑜 𝐵) ⊆ 1𝑜))
29	14, 28	mtod 189	. . . 4 ⊢ (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
30		elpri 4230	. . . . 5 ⊢ (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
31		df2o3 7618	. . . . 5 ⊢ 2𝑜 = {∅, 1𝑜}
32	30, 31	eleq2s 2748	. . . 4 ⊢ (𝐴 ∈ 2𝑜 → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
33	29, 32	nsyl 135	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 2𝑜)
34	1, 33	eldifd 3618	. 2 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 2𝑜))
35	34, 10	jca 553	1 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)))

Syntax hints:   → wi 4   ∨ wo 382   ∧ wa 383   = wceq 1523   ∈ wcel 2030  ∀wral 2941  ∃wrex 2942   ∖ cdif 3604   ⊆ wss 3607  ∅c0 3948  {cpr 4212  {copab 4745  dom cdm 5143  ran crn 5144  Oncon0 5761  ‘cfv 5926  (class class class)co 6690  1_𝑜c1o 7598  2_𝑜c2o 7599   ↑_𝑜 coe 7604   CNF ccnf 8596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-ord 5764  df-on 5765  df-lim 5766  df-suc 5767  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-ov 6693  df-oprab 6694  df-mpt2 6695  df-om 7108  df-wrecs 7452  df-recs 7513  df-rdg 7551  df-1o 7605  df-2o 7606  df-oadd 7609  df-omul 7610  df-oexp 7611

This theorem is referenced by:  cantnflem3  8626  cantnflem4  8627