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Theorem cantnflem2 8884
Description: Lemma for cantnf 8887. (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 7901 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
41, 2, 3syl2anc 579 . . . . . . . . 9 (𝜑 → (𝐴o 𝐵) ∈ On)
5 cantnf.c . . . . . . . . 9 (𝜑𝐶 ∈ (𝐴o 𝐵))
6 onelon 6001 . . . . . . . . 9 (((𝐴o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴o 𝐵)) → 𝐶 ∈ On)
74, 5, 6syl2anc 579 . . . . . . . 8 (𝜑𝐶 ∈ On)
8 cantnf.e . . . . . . . 8 (𝜑 → ∅ ∈ 𝐶)
9 ondif1 7865 . . . . . . . 8 (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
107, 8, 9sylanbrc 578 . . . . . . 7 (𝜑𝐶 ∈ (On ∖ 1o))
1110eldifbd 3805 . . . . . 6 (𝜑 → ¬ 𝐶 ∈ 1o)
12 ssel 3815 . . . . . . 7 ((𝐴o 𝐵) ⊆ 1o → (𝐶 ∈ (𝐴o 𝐵) → 𝐶 ∈ 1o))
135, 12syl5com 31 . . . . . 6 (𝜑 → ((𝐴o 𝐵) ⊆ 1o𝐶 ∈ 1o))
1411, 13mtod 190 . . . . 5 (𝜑 → ¬ (𝐴o 𝐵) ⊆ 1o)
15 oe0m 7882 . . . . . . . . 9 (𝐵 ∈ On → (∅ ↑o 𝐵) = (1o𝐵))
162, 15syl 17 . . . . . . . 8 (𝜑 → (∅ ↑o 𝐵) = (1o𝐵))
17 difss 3960 . . . . . . . 8 (1o𝐵) ⊆ 1o
1816, 17syl6eqss 3874 . . . . . . 7 (𝜑 → (∅ ↑o 𝐵) ⊆ 1o)
19 oveq1 6929 . . . . . . . 8 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
2019sseq1d 3851 . . . . . . 7 (𝐴 = ∅ → ((𝐴o 𝐵) ⊆ 1o ↔ (∅ ↑o 𝐵) ⊆ 1o))
2118, 20syl5ibrcom 239 . . . . . 6 (𝜑 → (𝐴 = ∅ → (𝐴o 𝐵) ⊆ 1o))
22 oe1m 7909 . . . . . . . 8 (𝐵 ∈ On → (1oo 𝐵) = 1o)
23 eqimss 3876 . . . . . . . 8 ((1oo 𝐵) = 1o → (1oo 𝐵) ⊆ 1o)
242, 22, 233syl 18 . . . . . . 7 (𝜑 → (1oo 𝐵) ⊆ 1o)
25 oveq1 6929 . . . . . . . 8 (𝐴 = 1o → (𝐴o 𝐵) = (1oo 𝐵))
2625sseq1d 3851 . . . . . . 7 (𝐴 = 1o → ((𝐴o 𝐵) ⊆ 1o ↔ (1oo 𝐵) ⊆ 1o))
2724, 26syl5ibrcom 239 . . . . . 6 (𝜑 → (𝐴 = 1o → (𝐴o 𝐵) ⊆ 1o))
2821, 27jaod 848 . . . . 5 (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1o) → (𝐴o 𝐵) ⊆ 1o))
2914, 28mtod 190 . . . 4 (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1o))
30 elpri 4420 . . . . 5 (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
31 df2o3 7857 . . . . 5 2o = {∅, 1o}
3230, 31eleq2s 2877 . . . 4 (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
3329, 32nsyl 138 . . 3 (𝜑 → ¬ 𝐴 ∈ 2o)
341, 33eldifd 3803 . 2 (𝜑𝐴 ∈ (On ∖ 2o))
3534, 10jca 507 1 (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 386  wo 836   = wceq 1601  wcel 2107  wral 3090  wrex 3091  cdif 3789  wss 3792  c0 4141  {cpr 4400  {copab 4948  dom cdm 5355  ran crn 5356  Oncon0 5976  cfv 6135  (class class class)co 6922  1oc1o 7836  2oc2o 7837  o coe 7842   CNF ccnf 8855
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-8 2109  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754  ax-rep 5006  ax-sep 5017  ax-nul 5025  ax-pow 5077  ax-pr 5138  ax-un 7226
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3or 1072  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-mo 2551  df-eu 2587  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ne 2970  df-ral 3095  df-rex 3096  df-reu 3097  df-rab 3099  df-v 3400  df-sbc 3653  df-csb 3752  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-pss 3808  df-nul 4142  df-if 4308  df-pw 4381  df-sn 4399  df-pr 4401  df-tp 4403  df-op 4405  df-uni 4672  df-iun 4755  df-br 4887  df-opab 4949  df-mpt 4966  df-tr 4988  df-id 5261  df-eprel 5266  df-po 5274  df-so 5275  df-fr 5314  df-we 5316  df-xp 5361  df-rel 5362  df-cnv 5363  df-co 5364  df-dm 5365  df-rn 5366  df-res 5367  df-ima 5368  df-pred 5933  df-ord 5979  df-on 5980  df-lim 5981  df-suc 5982  df-iota 6099  df-fun 6137  df-fn 6138  df-f 6139  df-f1 6140  df-fo 6141  df-f1o 6142  df-fv 6143  df-ov 6925  df-oprab 6926  df-mpt2 6927  df-om 7344  df-wrecs 7689  df-recs 7751  df-rdg 7789  df-1o 7843  df-2o 7844  df-oadd 7847  df-omul 7848  df-oexp 7849
This theorem is referenced by:  cantnflem3  8885  cantnflem4  8886
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