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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 (𝜑𝐴 ∈ On)
cantnfs.b (𝜑𝐵 ∈ On)
oemapval.t 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧𝐵 ((𝑥𝑧) ∈ (𝑦𝑧) ∧ ∀𝑤𝐵 (𝑧𝑤 → (𝑥𝑤) = (𝑦𝑤)))}
cantnf.c (𝜑𝐶 ∈ (𝐴𝑜 𝐵))
cantnf.s (𝜑𝐶 ⊆ 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
StepHypRef Expression
1 cantnfs.a . . 3 (𝜑𝐴 ∈ On)
2 cantnfs.b . . . . . . . . . 10 (𝜑𝐵 ∈ On)
3 oecl 7775 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴𝑜 𝐵) ∈ On)
41, 2, 3syl2anc 573 . . . . . . . . 9 (𝜑 → (𝐴𝑜 𝐵) ∈ On)
5 cantnf.c . . . . . . . . 9 (𝜑𝐶 ∈ (𝐴𝑜 𝐵))
6 onelon 5890 . . . . . . . . 9 (((𝐴𝑜 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴𝑜 𝐵)) → 𝐶 ∈ On)
74, 5, 6syl2anc 573 . . . . . . . 8 (𝜑𝐶 ∈ On)
8 cantnf.e . . . . . . . 8 (𝜑 → ∅ ∈ 𝐶)
9 ondif1 7739 . . . . . . . 8 (𝐶 ∈ (On ∖ 1𝑜) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
107, 8, 9sylanbrc 572 . . . . . . 7 (𝜑𝐶 ∈ (On ∖ 1𝑜))
1110eldifbd 3736 . . . . . 6 (𝜑 → ¬ 𝐶 ∈ 1𝑜)
12 ssel 3746 . . . . . . 7 ((𝐴𝑜 𝐵) ⊆ 1𝑜 → (𝐶 ∈ (𝐴𝑜 𝐵) → 𝐶 ∈ 1𝑜))
135, 12syl5com 31 . . . . . 6 (𝜑 → ((𝐴𝑜 𝐵) ⊆ 1𝑜𝐶 ∈ 1𝑜))
1411, 13mtod 189 . . . . 5 (𝜑 → ¬ (𝐴𝑜 𝐵) ⊆ 1𝑜)
15 oe0m 7756 . . . . . . . . 9 (𝐵 ∈ On → (∅ ↑𝑜 𝐵) = (1𝑜𝐵))
162, 15syl 17 . . . . . . . 8 (𝜑 → (∅ ↑𝑜 𝐵) = (1𝑜𝐵))
17 difss 3888 . . . . . . . 8 (1𝑜𝐵) ⊆ 1𝑜
1816, 17syl6eqss 3804 . . . . . . 7 (𝜑 → (∅ ↑𝑜 𝐵) ⊆ 1𝑜)
19 oveq1 6803 . . . . . . . 8 (𝐴 = ∅ → (𝐴𝑜 𝐵) = (∅ ↑𝑜 𝐵))
2019sseq1d 3781 . . . . . . 7 (𝐴 = ∅ → ((𝐴𝑜 𝐵) ⊆ 1𝑜 ↔ (∅ ↑𝑜 𝐵) ⊆ 1𝑜))
2118, 20syl5ibrcom 237 . . . . . 6 (𝜑 → (𝐴 = ∅ → (𝐴𝑜 𝐵) ⊆ 1𝑜))
22 oe1m 7783 . . . . . . . 8 (𝐵 ∈ On → (1𝑜𝑜 𝐵) = 1𝑜)
23 eqimss 3806 . . . . . . . 8 ((1𝑜𝑜 𝐵) = 1𝑜 → (1𝑜𝑜 𝐵) ⊆ 1𝑜)
242, 22, 233syl 18 . . . . . . 7 (𝜑 → (1𝑜𝑜 𝐵) ⊆ 1𝑜)
25 oveq1 6803 . . . . . . . 8 (𝐴 = 1𝑜 → (𝐴𝑜 𝐵) = (1𝑜𝑜 𝐵))
2625sseq1d 3781 . . . . . . 7 (𝐴 = 1𝑜 → ((𝐴𝑜 𝐵) ⊆ 1𝑜 ↔ (1𝑜𝑜 𝐵) ⊆ 1𝑜))
2724, 26syl5ibrcom 237 . . . . . 6 (𝜑 → (𝐴 = 1𝑜 → (𝐴𝑜 𝐵) ⊆ 1𝑜))
2821, 27jaod 848 . . . . 5 (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1𝑜) → (𝐴𝑜 𝐵) ⊆ 1𝑜))
2914, 28mtod 189 . . . 4 (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
30 elpri 4338 . . . . 5 (𝐴 ∈ {∅, 1𝑜} → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
31 df2o3 7731 . . . . 5 2𝑜 = {∅, 1𝑜}
3230, 31eleq2s 2868 . . . 4 (𝐴 ∈ 2𝑜 → (𝐴 = ∅ ∨ 𝐴 = 1𝑜))
3329, 32nsyl 137 . . 3 (𝜑 → ¬ 𝐴 ∈ 2𝑜)
341, 33eldifd 3734 . 2 (𝜑𝐴 ∈ (On ∖ 2𝑜))
3534, 10jca 501 1 (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜)))
Colors of variables: wff setvar class
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  1𝑜c1o 7710  2𝑜c2o 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
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