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Mirrors > Home > MPE Home > Th. List > cantnflem2 | Structured version Visualization version GIF version |
Description: Lemma for cantnf 8887. (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 7901 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On) | |
4 | 1, 2, 3 | syl2anc 579 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On) |
5 | cantnf.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵)) | |
6 | onelon 6001 | . . . . . . . . 9 ⊢ (((𝐴 ↑o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 ↑o 𝐵)) → 𝐶 ∈ On) | |
7 | 4, 5, 6 | syl2anc 579 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On) |
8 | cantnf.e | . . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐶) | |
9 | ondif1 7865 | . . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) | |
10 | 7, 8, 9 | sylanbrc 578 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (On ∖ 1o)) |
11 | 10 | eldifbd 3805 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ∈ 1o) |
12 | ssel 3815 | . . . . . . 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 190 | . . . . 5 ⊢ (𝜑 → ¬ (𝐴 ↑o 𝐵) ⊆ 1o) |
15 | oe0m 7882 | . . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) = (1o ∖ 𝐵)) | |
16 | 2, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (∅ ↑o 𝐵) = (1o ∖ 𝐵)) |
17 | difss 3960 | . . . . . . . 8 ⊢ (1o ∖ 𝐵) ⊆ 1o | |
18 | 16, 17 | syl6eqss 3874 | . . . . . . 7 ⊢ (𝜑 → (∅ ↑o 𝐵) ⊆ 1o) |
19 | oveq1 6929 | . . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵)) | |
20 | 19 | sseq1d 3851 | . . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ⊆ 1o ↔ (∅ ↑o 𝐵) ⊆ 1o)) |
21 | 18, 20 | syl5ibrcom 239 | . . . . . 6 ⊢ (𝜑 → (𝐴 = ∅ → (𝐴 ↑o 𝐵) ⊆ 1o)) |
22 | oe1m 7909 | . . . . . . . 8 ⊢ (𝐵 ∈ On → (1o ↑o 𝐵) = 1o) | |
23 | eqimss 3876 | . . . . . . . 8 ⊢ ((1o ↑o 𝐵) = 1o → (1o ↑o 𝐵) ⊆ 1o) | |
24 | 2, 22, 23 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → (1o ↑o 𝐵) ⊆ 1o) |
25 | oveq1 6929 | . . . . . . . 8 ⊢ (𝐴 = 1o → (𝐴 ↑o 𝐵) = (1o ↑o 𝐵)) | |
26 | 25 | sseq1d 3851 | . . . . . . 7 ⊢ (𝐴 = 1o → ((𝐴 ↑o 𝐵) ⊆ 1o ↔ (1o ↑o 𝐵) ⊆ 1o)) |
27 | 24, 26 | syl5ibrcom 239 | . . . . . 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 190 | . . . 4 ⊢ (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1o)) |
30 | elpri 4420 | . . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o)) | |
31 | df2o3 7857 | . . . . 5 ⊢ 2o = {∅, 1o} | |
32 | 30, 31 | eleq2s 2877 | . . . 4 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o)) |
33 | 29, 32 | nsyl 138 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 2o) |
34 | 1, 33 | eldifd 3803 | . 2 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 2o)) |
35 | 34, 10 | jca 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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