Theorem cantnflem2 9448

Description: Lemma for cantnf 9451. (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 8367	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
4	1, 2, 3	syl2anc 584	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On)
5		cantnf.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))
6		onelon 6291	. . . . . . . . 9 ⊢ (((𝐴 ↑o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 ↑o 𝐵)) → 𝐶 ∈ On)
7	4, 5, 6	syl2anc 584	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On)
8		cantnf.e	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐶)
9		ondif1 8331	. . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
10	7, 8, 9	sylanbrc 583	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (On ∖ 1o))
11	10	eldifbd 3900	. . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ∈ 1o)
12		ssel 3914	. . . . . . 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 197	. . . . 5 ⊢ (𝜑 → ¬ (𝐴 ↑o 𝐵) ⊆ 1o)
15		oe0m 8348	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) = (1o ∖ 𝐵))
16	2, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → (∅ ↑o 𝐵) = (1o ∖ 𝐵))
17		difss 4066	. . . . . . . 8 ⊢ (1o ∖ 𝐵) ⊆ 1o
18	16, 17	eqsstrdi 3975	. . . . . . 7 ⊢ (𝜑 → (∅ ↑o 𝐵) ⊆ 1o)
19		oveq1 7282	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
20	19	sseq1d 3952	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ⊆ 1o ↔ (∅ ↑o 𝐵) ⊆ 1o))
21	18, 20	syl5ibrcom 246	. . . . . 6 ⊢ (𝜑 → (𝐴 = ∅ → (𝐴 ↑o 𝐵) ⊆ 1o))
22		oe1m 8376	. . . . . . . 8 ⊢ (𝐵 ∈ On → (1o ↑o 𝐵) = 1o)
23		eqimss 3977	. . . . . . . 8 ⊢ ((1o ↑o 𝐵) = 1o → (1o ↑o 𝐵) ⊆ 1o)
24	2, 22, 23	3syl 18	. . . . . . 7 ⊢ (𝜑 → (1o ↑o 𝐵) ⊆ 1o)
25		oveq1 7282	. . . . . . . 8 ⊢ (𝐴 = 1o → (𝐴 ↑o 𝐵) = (1o ↑o 𝐵))
26	25	sseq1d 3952	. . . . . . 7 ⊢ (𝐴 = 1o → ((𝐴 ↑o 𝐵) ⊆ 1o ↔ (1o ↑o 𝐵) ⊆ 1o))
27	24, 26	syl5ibrcom 246	. . . . . 6 ⊢ (𝜑 → (𝐴 = 1o → (𝐴 ↑o 𝐵) ⊆ 1o))
28	21, 27	jaod 856	. . . . 5 ⊢ (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1o) → (𝐴 ↑o 𝐵) ⊆ 1o))
29	14, 28	mtod 197	. . . 4 ⊢ (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1o))
30		elpri 4583	. . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
31		df2o3 8305	. . . . 5 ⊢ 2o = {∅, 1o}
32	30, 31	eleq2s 2857	. . . 4 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
33	29, 32	nsyl 140	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 2o)
34	1, 33	eldifd 3898	. 2 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 2o))
35	34, 10	jca 512	1 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))

Syntax hints:   → wi 4   ∧ wa 396   ∨ wo 844   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ∖ cdif 3884   ⊆ wss 3887  ∅c0 4256  {cpr 4563  {copab 5136  dom cdm 5589  ran crn 5590  Oncon0 6266  ‘cfv 6433  (class class class)co 7275  1_oc1o 8290  2_oc2o 8291   ↑_o coe 8296   CNF ccnf 9419

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-rep 5209  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-reu 3072  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-tr 5192  df-id 5489  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-pred 6202  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-ov 7278  df-oprab 7279  df-mpo 7280  df-om 7713  df-2nd 7832  df-frecs 8097  df-wrecs 8128  df-recs 8202  df-rdg 8241  df-1o 8297  df-2o 8298  df-oadd 8301  df-omul 8302  df-oexp 8303

This theorem is referenced by:  cantnflem3  9449  cantnflem4  9450