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Mirrors > Home > MPE Home > Th. List > cantnflem2 | Structured version Visualization version GIF version |
Description: Lemma for cantnf 9690. (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 8539 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On) | |
4 | 1, 2, 3 | syl2anc 582 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On) |
5 | cantnf.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵)) | |
6 | onelon 6388 | . . . . . . . . 9 ⊢ (((𝐴 ↑o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 ↑o 𝐵)) → 𝐶 ∈ On) | |
7 | 4, 5, 6 | syl2anc 582 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On) |
8 | cantnf.e | . . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐶) | |
9 | ondif1 8503 | . . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) | |
10 | 7, 8, 9 | sylanbrc 581 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (On ∖ 1o)) |
11 | 10 | eldifbd 3960 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ∈ 1o) |
12 | ssel 3974 | . . . . . . 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 197 | . . . . 5 ⊢ (𝜑 → ¬ (𝐴 ↑o 𝐵) ⊆ 1o) |
15 | oe0m 8520 | . . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) = (1o ∖ 𝐵)) | |
16 | 2, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (∅ ↑o 𝐵) = (1o ∖ 𝐵)) |
17 | difss 4130 | . . . . . . . 8 ⊢ (1o ∖ 𝐵) ⊆ 1o | |
18 | 16, 17 | eqsstrdi 4035 | . . . . . . 7 ⊢ (𝜑 → (∅ ↑o 𝐵) ⊆ 1o) |
19 | oveq1 7418 | . . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵)) | |
20 | 19 | sseq1d 4012 | . . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ⊆ 1o ↔ (∅ ↑o 𝐵) ⊆ 1o)) |
21 | 18, 20 | syl5ibrcom 246 | . . . . . 6 ⊢ (𝜑 → (𝐴 = ∅ → (𝐴 ↑o 𝐵) ⊆ 1o)) |
22 | oe1m 8547 | . . . . . . . 8 ⊢ (𝐵 ∈ On → (1o ↑o 𝐵) = 1o) | |
23 | eqimss 4039 | . . . . . . . 8 ⊢ ((1o ↑o 𝐵) = 1o → (1o ↑o 𝐵) ⊆ 1o) | |
24 | 2, 22, 23 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → (1o ↑o 𝐵) ⊆ 1o) |
25 | oveq1 7418 | . . . . . . . 8 ⊢ (𝐴 = 1o → (𝐴 ↑o 𝐵) = (1o ↑o 𝐵)) | |
26 | 25 | sseq1d 4012 | . . . . . . 7 ⊢ (𝐴 = 1o → ((𝐴 ↑o 𝐵) ⊆ 1o ↔ (1o ↑o 𝐵) ⊆ 1o)) |
27 | 24, 26 | syl5ibrcom 246 | . . . . . 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 197 | . . . 4 ⊢ (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1o)) |
30 | elpri 4649 | . . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o)) | |
31 | df2o3 8476 | . . . . 5 ⊢ 2o = {∅, 1o} | |
32 | 30, 31 | eleq2s 2849 | . . . 4 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o)) |
33 | 29, 32 | nsyl 140 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 2o) |
34 | 1, 33 | eldifd 3958 | . 2 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 2o)) |
35 | 34, 10 | jca 510 | 1 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∨ wo 843 = wceq 1539 ∈ wcel 2104 ∀wral 3059 ∃wrex 3068 ∖ cdif 3944 ⊆ wss 3947 ∅c0 4321 {cpr 4629 {copab 5209 dom cdm 5675 ran crn 5676 Oncon0 6363 ‘cfv 6542 (class class class)co 7411 1oc1o 8461 2oc2o 8462 ↑o coe 8467 CNF ccnf 9658 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 ax-un 7727 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-pss 3966 df-nul 4322 df-if 4528 df-pw 4603 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-tr 5265 df-id 5573 df-eprel 5579 df-po 5587 df-so 5588 df-fr 5630 df-we 5632 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-pred 6299 df-ord 6366 df-on 6367 df-lim 6368 df-suc 6369 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-ov 7414 df-oprab 7415 df-mpo 7416 df-om 7858 df-2nd 7978 df-frecs 8268 df-wrecs 8299 df-recs 8373 df-rdg 8412 df-1o 8468 df-2o 8469 df-oadd 8472 df-omul 8473 df-oexp 8474 |
This theorem is referenced by: cantnflem3 9688 cantnflem4 9689 |
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