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Mirrors > Home > MPE Home > Th. List > cantnflem2 | Structured version Visualization version GIF version |
Description: Lemma for cantnf 8758. (Contributed by Mario Carneiro, 28-May-2015.) |
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) |
Ref | Expression |
---|---|
cantnflem2 | ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2𝑜) ∧ 𝐶 ∈ (On ∖ 1𝑜))) |
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))) |
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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