Theorem cantnflem2 8755

Description: Lemma for cantnf 8758. (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 7775	. . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑𝑜 𝐵) ∈ On)
4	1, 2, 3	syl2anc 573	. . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑𝑜 𝐵) ∈ On)
5		cantnf.c	. . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑𝑜 𝐵))
6		onelon 5890	. . . . . . . . 9 ⊢ (((𝐴 ↑𝑜 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 ↑𝑜 𝐵)) → 𝐶 ∈ On)
7	4, 5, 6	syl2anc 573	. . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On)
8		cantnf.e	. . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐶)
9		ondif1 7739	. . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 1𝑜) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
10	7, 8, 9	sylanbrc 572	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (On ∖ 1𝑜))
11	10	eldifbd 3736	. . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ∈ 1𝑜)
12		ssel 3746	. . . . . . 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 7756	. . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ↑𝑜 𝐵) = (1𝑜 ∖ 𝐵))
16	2, 15	syl 17	. . . . . . . 8 ⊢ (𝜑 → (∅ ↑𝑜 𝐵) = (1𝑜 ∖ 𝐵))
17		difss 3888	. . . . . . . 8 ⊢ (1𝑜 ∖ 𝐵) ⊆ 1𝑜
18	16, 17	syl6eqss 3804	. . . . . . 7 ⊢ (𝜑 → (∅ ↑𝑜 𝐵) ⊆ 1𝑜)
19		oveq1 6803	. . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) = (∅ ↑𝑜 𝐵))
20	19	sseq1d 3781	. . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐴 ↑𝑜 𝐵) ⊆ 1𝑜 ↔ (∅ ↑𝑜 𝐵) ⊆ 1𝑜))
21	18, 20	syl5ibrcom 237	. . . . . 6 ⊢ (𝜑 → (𝐴 = ∅ → (𝐴 ↑𝑜 𝐵) ⊆ 1𝑜))
22		oe1m 7783	. . . . . . . 8 ⊢ (𝐵 ∈ On → (1𝑜 ↑𝑜 𝐵) = 1𝑜)
23		eqimss 3806	. . . . . . . 8 ⊢ ((1𝑜 ↑𝑜 𝐵) = 1𝑜 → (1𝑜 ↑𝑜 𝐵) ⊆ 1𝑜)
24	2, 22, 23	3syl 18	. . . . . . 7 ⊢ (𝜑 → (1𝑜 ↑𝑜 𝐵) ⊆ 1𝑜)
25		oveq1 6803	. . . . . . . 8 ⊢ (𝐴 = 1𝑜 → (𝐴 ↑𝑜 𝐵) = (1𝑜 ↑𝑜 𝐵))
26	25	sseq1d 3781	. . . . . . 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 848	. . . . 5 ⊢ (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → (𝐴 ↑𝑜 𝐵) ⊆ 1𝑜))
29	14, 28	mtod 189	. . . 4 ⊢ (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
30		elpri 4338	. . . . 5 ⊢ (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
31		df2o3 7731	. . . . 5 ⊢ 2𝑜 = {∅, 1𝑜}
32	30, 31	eleq2s 2868	. . . 4 ⊢ (𝐴 ∈ 2𝑜 → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
33	29, 32	nsyl 137	. . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 2𝑜)
34	1, 33	eldifd 3734	. 2 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 2𝑜))
35	34, 10	jca 501	1 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)))

Syntax hints:   → wi 4   ∧ wa 382   ∨ wo 836   = wceq 1631   ∈ wcel 2145  ∀wral 3061  ∃wrex 3062   ∖ cdif 3720   ⊆ wss 3723  ∅c0 4063  {cpr 4319  {copab 4847  dom cdm 5250  ran crn 5251  Oncon0 5865  ‘cfv 6030  (class class class)co 6796  1_𝑜c1o 7710  2_𝑜c2o 7711   ↑_𝑜 coe 7716   CNF ccnf 8726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4905  ax-sep 4916  ax-nul 4924  ax-pow 4975  ax-pr 5035  ax-un 7100

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4227  df-pw 4300  df-sn 4318  df-pr 4320  df-tp 4322  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-tr 4888  df-id 5158  df-eprel 5163  df-po 5171  df-so 5172  df-fr 5209  df-we 5211  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-pred 5822  df-ord 5868  df-on 5869  df-lim 5870  df-suc 5871  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-f1 6035  df-fo 6036  df-f1o 6037  df-fv 6038  df-ov 6799  df-oprab 6800  df-mpt2 6801  df-om 7217  df-wrecs 7563  df-recs 7625  df-rdg 7663  df-1o 7717  df-2o 7718  df-oadd 7721  df-omul 7722  df-oexp 7723

This theorem is referenced by:  cantnflem3  8756  cantnflem4  8757