Theorem cantnflem2 9688

Description: Lemma for cantnf 9691. (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 8540	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On)
4	1, 2, 3	syl2anc 583	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On)
5		cantnf.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵))
6		onelon 6390	. . . . . . . . 9 ⊢ (((𝐴 ↑o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 ↑o 𝐵)) → 𝐶 ∈ On)
7	4, 5, 6	syl2anc 583	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On)
8		cantnf.e	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐶)
9		ondif1 8504	. . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
10	7, 8, 9	sylanbrc 582	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (On ∖ 1o))
11	10	eldifbd 3962	. . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ∈ 1o)
12		ssel 3976	. . . . . . 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 8521	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) = (1o ∖ 𝐵))
16	2, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → (∅ ↑o 𝐵) = (1o ∖ 𝐵))
17		difss 4132	. . . . . . . 8 ⊢ (1o ∖ 𝐵) ⊆ 1o
18	16, 17	eqsstrdi 4037	. . . . . . 7 ⊢ (𝜑 → (∅ ↑o 𝐵) ⊆ 1o)
19		oveq1 7419	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵))
20	19	sseq1d 4014	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ⊆ 1o ↔ (∅ ↑o 𝐵) ⊆ 1o))
21	18, 20	syl5ibrcom 246	. . . . . 6 ⊢ (𝜑 → (𝐴 = ∅ → (𝐴 ↑o 𝐵) ⊆ 1o))
22		oe1m 8548	. . . . . . . 8 ⊢ (𝐵 ∈ On → (1o ↑o 𝐵) = 1o)
23		eqimss 4041	. . . . . . . 8 ⊢ ((1o ↑o 𝐵) = 1o → (1o ↑o 𝐵) ⊆ 1o)
24	2, 22, 23	3syl 18	. . . . . . 7 ⊢ (𝜑 → (1o ↑o 𝐵) ⊆ 1o)
25		oveq1 7419	. . . . . . . 8 ⊢ (𝐴 = 1o → (𝐴 ↑o 𝐵) = (1o ↑o 𝐵))
26	25	sseq1d 4014	. . . . . . 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 4651	. . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
31		df2o3 8477	. . . . 5 ⊢ 2o = {∅, 1o}
32	30, 31	eleq2s 2850	. . . 4 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
33	29, 32	nsyl 140	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 2o)
34	1, 33	eldifd 3960	. 2 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 2o))
35	34, 10	jca 511	1 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))

Syntax hints:   → wi 4   ∧ wa 395   ∨ wo 844   = wceq 1540   ∈ wcel 2105  ∀wral 3060  ∃wrex 3069   ∖ cdif 3946   ⊆ wss 3949  ∅c0 4323  {cpr 4631  {copab 5211  dom cdm 5677  ran crn 5678  Oncon0 6365  ‘cfv 6544  (class class class)co 7412  1_oc1o 8462  2_oc2o 8463   ↑_o coe 8468   CNF ccnf 9659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2702  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7728

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-reu 3376  df-rab 3432  df-v 3475  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-pred 6301  df-ord 6368  df-on 6369  df-lim 6370  df-suc 6371  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-ov 7415  df-oprab 7416  df-mpo 7417  df-om 7859  df-2nd 7979  df-frecs 8269  df-wrecs 8300  df-recs 8374  df-rdg 8413  df-1o 8469  df-2o 8470  df-oadd 8473  df-omul 8474  df-oexp 8475

This theorem is referenced by:  cantnflem3  9689  cantnflem4  9690