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Theorem cantnflem2 9634
Description: Lemma for cantnf 9637. (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 8487 . . . . . . . . . 10 ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴o 𝐵) ∈ On)
41, 2, 3syl2anc 585 . . . . . . . . 9 (𝜑 → (𝐴o 𝐵) ∈ On)
5 cantnf.c . . . . . . . . 9 (𝜑𝐶 ∈ (𝐴o 𝐵))
6 onelon 6346 . . . . . . . . 9 (((𝐴o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴o 𝐵)) → 𝐶 ∈ On)
74, 5, 6syl2anc 585 . . . . . . . 8 (𝜑𝐶 ∈ On)
8 cantnf.e . . . . . . . 8 (𝜑 → ∅ ∈ 𝐶)
9 ondif1 8451 . . . . . . . 8 (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶))
107, 8, 9sylanbrc 584 . . . . . . 7 (𝜑𝐶 ∈ (On ∖ 1o))
1110eldifbd 3927 . . . . . 6 (𝜑 → ¬ 𝐶 ∈ 1o)
12 ssel 3941 . . . . . . 7 ((𝐴o 𝐵) ⊆ 1o → (𝐶 ∈ (𝐴o 𝐵) → 𝐶 ∈ 1o))
135, 12syl5com 31 . . . . . 6 (𝜑 → ((𝐴o 𝐵) ⊆ 1o𝐶 ∈ 1o))
1411, 13mtod 197 . . . . 5 (𝜑 → ¬ (𝐴o 𝐵) ⊆ 1o)
15 oe0m 8468 . . . . . . . . 9 (𝐵 ∈ On → (∅ ↑o 𝐵) = (1o𝐵))
162, 15syl 17 . . . . . . . 8 (𝜑 → (∅ ↑o 𝐵) = (1o𝐵))
17 difss 4095 . . . . . . . 8 (1o𝐵) ⊆ 1o
1816, 17eqsstrdi 4002 . . . . . . 7 (𝜑 → (∅ ↑o 𝐵) ⊆ 1o)
19 oveq1 7368 . . . . . . . 8 (𝐴 = ∅ → (𝐴o 𝐵) = (∅ ↑o 𝐵))
2019sseq1d 3979 . . . . . . 7 (𝐴 = ∅ → ((𝐴o 𝐵) ⊆ 1o ↔ (∅ ↑o 𝐵) ⊆ 1o))
2118, 20syl5ibrcom 247 . . . . . 6 (𝜑 → (𝐴 = ∅ → (𝐴o 𝐵) ⊆ 1o))
22 oe1m 8496 . . . . . . . 8 (𝐵 ∈ On → (1oo 𝐵) = 1o)
23 eqimss 4004 . . . . . . . 8 ((1oo 𝐵) = 1o → (1oo 𝐵) ⊆ 1o)
242, 22, 233syl 18 . . . . . . 7 (𝜑 → (1oo 𝐵) ⊆ 1o)
25 oveq1 7368 . . . . . . . 8 (𝐴 = 1o → (𝐴o 𝐵) = (1oo 𝐵))
2625sseq1d 3979 . . . . . . 7 (𝐴 = 1o → ((𝐴o 𝐵) ⊆ 1o ↔ (1oo 𝐵) ⊆ 1o))
2724, 26syl5ibrcom 247 . . . . . 6 (𝜑 → (𝐴 = 1o → (𝐴o 𝐵) ⊆ 1o))
2821, 27jaod 858 . . . . 5 (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1o) → (𝐴o 𝐵) ⊆ 1o))
2914, 28mtod 197 . . . 4 (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1o))
30 elpri 4612 . . . . 5 (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o))
31 df2o3 8424 . . . . 5 2o = {∅, 1o}
3230, 31eleq2s 2852 . . . 4 (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o))
3329, 32nsyl 140 . . 3 (𝜑 → ¬ 𝐴 ∈ 2o)
341, 33eldifd 3925 . 2 (𝜑𝐴 ∈ (On ∖ 2o))
3534, 10jca 513 1 (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  wo 846   = wceq 1542  wcel 2107  wral 3061  wrex 3070  cdif 3911  wss 3914  c0 4286  {cpr 4592  {copab 5171  dom cdm 5637  ran crn 5638  Oncon0 6321  cfv 6500  (class class class)co 7361  1oc1o 8409  2oc2o 8410  o coe 8415   CNF ccnf 9605
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-oprab 7365  df-mpo 7366  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360  df-1o 8416  df-2o 8417  df-oadd 8420  df-omul 8421  df-oexp 8422
This theorem is referenced by:  cantnflem3  9635  cantnflem4  9636
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