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Mirrors > Home > MPE Home > Th. List > cantnflem2 | Structured version Visualization version GIF version |
Description: Lemma for cantnf 9150. (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 8156 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On) | |
4 | 1, 2, 3 | syl2anc 586 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On) |
5 | cantnf.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵)) | |
6 | onelon 6211 | . . . . . . . . 9 ⊢ (((𝐴 ↑o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 ↑o 𝐵)) → 𝐶 ∈ On) | |
7 | 4, 5, 6 | syl2anc 586 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On) |
8 | cantnf.e | . . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐶) | |
9 | ondif1 8120 | . . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) | |
10 | 7, 8, 9 | sylanbrc 585 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (On ∖ 1o)) |
11 | 10 | eldifbd 3949 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ∈ 1o) |
12 | ssel 3961 | . . . . . . 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 200 | . . . . 5 ⊢ (𝜑 → ¬ (𝐴 ↑o 𝐵) ⊆ 1o) |
15 | oe0m 8137 | . . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) = (1o ∖ 𝐵)) | |
16 | 2, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (∅ ↑o 𝐵) = (1o ∖ 𝐵)) |
17 | difss 4108 | . . . . . . . 8 ⊢ (1o ∖ 𝐵) ⊆ 1o | |
18 | 16, 17 | eqsstrdi 4021 | . . . . . . 7 ⊢ (𝜑 → (∅ ↑o 𝐵) ⊆ 1o) |
19 | oveq1 7157 | . . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵)) | |
20 | 19 | sseq1d 3998 | . . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ⊆ 1o ↔ (∅ ↑o 𝐵) ⊆ 1o)) |
21 | 18, 20 | syl5ibrcom 249 | . . . . . 6 ⊢ (𝜑 → (𝐴 = ∅ → (𝐴 ↑o 𝐵) ⊆ 1o)) |
22 | oe1m 8165 | . . . . . . . 8 ⊢ (𝐵 ∈ On → (1o ↑o 𝐵) = 1o) | |
23 | eqimss 4023 | . . . . . . . 8 ⊢ ((1o ↑o 𝐵) = 1o → (1o ↑o 𝐵) ⊆ 1o) | |
24 | 2, 22, 23 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → (1o ↑o 𝐵) ⊆ 1o) |
25 | oveq1 7157 | . . . . . . . 8 ⊢ (𝐴 = 1o → (𝐴 ↑o 𝐵) = (1o ↑o 𝐵)) | |
26 | 25 | sseq1d 3998 | . . . . . . 7 ⊢ (𝐴 = 1o → ((𝐴 ↑o 𝐵) ⊆ 1o ↔ (1o ↑o 𝐵) ⊆ 1o)) |
27 | 24, 26 | syl5ibrcom 249 | . . . . . 6 ⊢ (𝜑 → (𝐴 = 1o → (𝐴 ↑o 𝐵) ⊆ 1o)) |
28 | 21, 27 | jaod 855 | . . . . 5 ⊢ (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1o) → (𝐴 ↑o 𝐵) ⊆ 1o)) |
29 | 14, 28 | mtod 200 | . . . 4 ⊢ (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1o)) |
30 | elpri 4583 | . . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o)) | |
31 | df2o3 8111 | . . . . 5 ⊢ 2o = {∅, 1o} | |
32 | 30, 31 | eleq2s 2931 | . . . 4 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o)) |
33 | 29, 32 | nsyl 142 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 2o) |
34 | 1, 33 | eldifd 3947 | . 2 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 2o)) |
35 | 34, 10 | jca 514 | 1 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∨ wo 843 = wceq 1533 ∈ wcel 2110 ∀wral 3138 ∃wrex 3139 ∖ cdif 3933 ⊆ wss 3936 ∅c0 4291 {cpr 4563 {copab 5121 dom cdm 5550 ran crn 5551 Oncon0 6186 ‘cfv 6350 (class class class)co 7150 1oc1o 8089 2oc2o 8090 ↑o coe 8095 CNF ccnf 9118 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2156 ax-12 2172 ax-ext 2793 ax-rep 5183 ax-sep 5196 ax-nul 5203 ax-pow 5259 ax-pr 5322 ax-un 7455 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3497 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4562 df-pr 4564 df-tp 4566 df-op 4568 df-uni 4833 df-iun 4914 df-br 5060 df-opab 5122 df-mpt 5140 df-tr 5166 df-id 5455 df-eprel 5460 df-po 5469 df-so 5470 df-fr 5509 df-we 5511 df-xp 5556 df-rel 5557 df-cnv 5558 df-co 5559 df-dm 5560 df-rn 5561 df-res 5562 df-ima 5563 df-pred 6143 df-ord 6189 df-on 6190 df-lim 6191 df-suc 6192 df-iota 6309 df-fun 6352 df-fn 6353 df-f 6354 df-f1 6355 df-fo 6356 df-f1o 6357 df-fv 6358 df-ov 7153 df-oprab 7154 df-mpo 7155 df-om 7575 df-wrecs 7941 df-recs 8002 df-rdg 8040 df-1o 8096 df-2o 8097 df-oadd 8100 df-omul 8101 df-oexp 8102 |
This theorem is referenced by: cantnflem3 9148 cantnflem4 9149 |
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