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Mirrors > Home > MPE Home > Th. List > cantnflem2 | Structured version Visualization version GIF version |
Description: Lemma for cantnf 9321. (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 8273 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On) | |
4 | 1, 2, 3 | syl2anc 587 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On) |
5 | cantnf.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵)) | |
6 | onelon 6247 | . . . . . . . . 9 ⊢ (((𝐴 ↑o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 ↑o 𝐵)) → 𝐶 ∈ On) | |
7 | 4, 5, 6 | syl2anc 587 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On) |
8 | cantnf.e | . . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐶) | |
9 | ondif1 8237 | . . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) | |
10 | 7, 8, 9 | sylanbrc 586 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (On ∖ 1o)) |
11 | 10 | eldifbd 3888 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ∈ 1o) |
12 | ssel 3902 | . . . . . . 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 201 | . . . . 5 ⊢ (𝜑 → ¬ (𝐴 ↑o 𝐵) ⊆ 1o) |
15 | oe0m 8254 | . . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) = (1o ∖ 𝐵)) | |
16 | 2, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (∅ ↑o 𝐵) = (1o ∖ 𝐵)) |
17 | difss 4055 | . . . . . . . 8 ⊢ (1o ∖ 𝐵) ⊆ 1o | |
18 | 16, 17 | eqsstrdi 3964 | . . . . . . 7 ⊢ (𝜑 → (∅ ↑o 𝐵) ⊆ 1o) |
19 | oveq1 7229 | . . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵)) | |
20 | 19 | sseq1d 3941 | . . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ⊆ 1o ↔ (∅ ↑o 𝐵) ⊆ 1o)) |
21 | 18, 20 | syl5ibrcom 250 | . . . . . 6 ⊢ (𝜑 → (𝐴 = ∅ → (𝐴 ↑o 𝐵) ⊆ 1o)) |
22 | oe1m 8282 | . . . . . . . 8 ⊢ (𝐵 ∈ On → (1o ↑o 𝐵) = 1o) | |
23 | eqimss 3966 | . . . . . . . 8 ⊢ ((1o ↑o 𝐵) = 1o → (1o ↑o 𝐵) ⊆ 1o) | |
24 | 2, 22, 23 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → (1o ↑o 𝐵) ⊆ 1o) |
25 | oveq1 7229 | . . . . . . . 8 ⊢ (𝐴 = 1o → (𝐴 ↑o 𝐵) = (1o ↑o 𝐵)) | |
26 | 25 | sseq1d 3941 | . . . . . . 7 ⊢ (𝐴 = 1o → ((𝐴 ↑o 𝐵) ⊆ 1o ↔ (1o ↑o 𝐵) ⊆ 1o)) |
27 | 24, 26 | syl5ibrcom 250 | . . . . . 6 ⊢ (𝜑 → (𝐴 = 1o → (𝐴 ↑o 𝐵) ⊆ 1o)) |
28 | 21, 27 | jaod 859 | . . . . 5 ⊢ (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1o) → (𝐴 ↑o 𝐵) ⊆ 1o)) |
29 | 14, 28 | mtod 201 | . . . 4 ⊢ (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1o)) |
30 | elpri 4572 | . . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o)) | |
31 | df2o3 8226 | . . . . 5 ⊢ 2o = {∅, 1o} | |
32 | 30, 31 | eleq2s 2857 | . . . 4 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o)) |
33 | 29, 32 | nsyl 142 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 2o) |
34 | 1, 33 | eldifd 3886 | . 2 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 2o)) |
35 | 34, 10 | jca 515 | 1 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2111 ∀wral 3062 ∃wrex 3063 ∖ cdif 3872 ⊆ wss 3875 ∅c0 4246 {cpr 4552 {copab 5124 dom cdm 5560 ran crn 5561 Oncon0 6222 ‘cfv 6389 (class class class)co 7222 1oc1o 8204 2oc2o 8205 ↑o coe 8210 CNF ccnf 9289 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-rep 5188 ax-sep 5201 ax-nul 5208 ax-pr 5331 ax-un 7532 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-reu 3069 df-rab 3071 df-v 3417 df-sbc 3704 df-csb 3821 df-dif 3878 df-un 3880 df-in 3882 df-ss 3892 df-pss 3894 df-nul 4247 df-if 4449 df-pw 4524 df-sn 4551 df-pr 4553 df-tp 4555 df-op 4557 df-uni 4829 df-iun 4915 df-br 5063 df-opab 5125 df-mpt 5145 df-tr 5171 df-id 5464 df-eprel 5469 df-po 5477 df-so 5478 df-fr 5518 df-we 5520 df-xp 5566 df-rel 5567 df-cnv 5568 df-co 5569 df-dm 5570 df-rn 5571 df-res 5572 df-ima 5573 df-pred 6169 df-ord 6225 df-on 6226 df-lim 6227 df-suc 6228 df-iota 6347 df-fun 6391 df-fn 6392 df-f 6393 df-f1 6394 df-fo 6395 df-f1o 6396 df-fv 6397 df-ov 7225 df-oprab 7226 df-mpo 7227 df-om 7654 df-wrecs 8056 df-recs 8117 df-rdg 8155 df-1o 8211 df-2o 8212 df-oadd 8215 df-omul 8216 df-oexp 8217 |
This theorem is referenced by: cantnflem3 9319 cantnflem4 9320 |
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