Theorem cantnflem2 8587

Description: Lemma for cantnf 8590. (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 7617	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)
4	1, 2, 3	syl2anc 693	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑𝑜 𝐵) ∈ On)
5		cantnf.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑𝑜 𝐵))
6		onelon 5748	. . . . . . . . 9 ⊢ (((𝐴 ↑𝑜 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 ↑𝑜 𝐵)) → 𝐶 ∈ On)
7	4, 5, 6	syl2anc 693	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On)
8		cantnf.e	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐶)
9		ondif1 7581	. . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 1𝑜) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
10	7, 8, 9	sylanbrc 698	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (On ∖ 1𝑜))
11	10	eldifbd 3587	. . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ∈ 1𝑜)
12		ssel 3597	. . . . . . 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 7598	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ↑𝑜 𝐵) = (1𝑜 ∖ 𝐵))
16	2, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → (∅ ↑𝑜 𝐵) = (1𝑜 ∖ 𝐵))
17		difss 3737	. . . . . . . 8 ⊢ (1𝑜 ∖ 𝐵) ⊆ 1𝑜
18	16, 17	syl6eqss 3655	. . . . . . 7 ⊢ (𝜑 → (∅ ↑𝑜 𝐵) ⊆ 1𝑜)
19		oveq1 6657	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 𝐵))
20	19	sseq1d 3632	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐴 ↑𝑜 𝐵) ⊆ 1𝑜 ↔ (∅ ↑𝑜 𝐵) ⊆ 1𝑜))
21	18, 20	syl5ibrcom 237	. . . . . 6 ⊢ (𝜑 → (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) ⊆ 1𝑜))
22		oe1m 7625	. . . . . . . 8 ⊢ (𝐵 ∈ On → (1𝑜 ↑𝑜 𝐵) = 1𝑜)
23		eqimss 3657	. . . . . . . 8 ⊢ ((1𝑜 ↑𝑜 𝐵) = 1𝑜 → (1𝑜 ↑𝑜 𝐵) ⊆ 1𝑜)
24	2, 22, 23	3syl 18	. . . . . . 7 ⊢ (𝜑 → (1𝑜 ↑𝑜 𝐵) ⊆ 1𝑜)
25		oveq1 6657	. . . . . . . 8 ⊢ (𝐴 = 1𝑜 → (𝐴 ↑𝑜 𝐵) = (1𝑜 ↑𝑜 𝐵))
26	25	sseq1d 3632	. . . . . . 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 395	. . . . 5 ⊢ (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → (𝐴 ↑𝑜 𝐵) ⊆ 1𝑜))
29	14, 28	mtod 189	. . . 4 ⊢ (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
30		elpri 4197	. . . . 5 ⊢ (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
31		df2o3 7573	. . . . 5 ⊢ 2𝑜 = {∅, 1𝑜}
32	30, 31	eleq2s 2719	. . . 4 ⊢ (𝐴 ∈ 2𝑜 → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
33	29, 32	nsyl 135	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 2𝑜)
34	1, 33	eldifd 3585	. 2 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 2𝑜))
35	34, 10	jca 554	1 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)))

Syntax hints:   → wi 4   ∨ wo 383   ∧ wa 384   = wceq 1483   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ∖ cdif 3571   ⊆ wss 3574  ∅c0 3915  {cpr 4179  {copab 4712  dom cdm 5114  ran crn 5115  Oncon0 5723  ‘cfv 5888  (class class class)co 6650  1_𝑜c1o 7553  2_𝑜c2o 7554   ↑_𝑜 coe 7559   CNF ccnf 8558

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-1o 7560  df-2o 7561  df-oadd 7564  df-omul 7565  df-oexp 7566

This theorem is referenced by:  cantnflem3  8588  cantnflem4  8589