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| Mirrors > Home > MPE Home > Th. List > cantnflem2 | Structured version Visualization version GIF version | ||
| Description: Lemma for cantnf 9733. (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 8575 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (𝐴 ↑o 𝐵) ∈ On) | |
| 4 | 1, 2, 3 | syl2anc 584 | . . . . . . . . 9 ⊢ (𝜑 → (𝐴 ↑o 𝐵) ∈ On) |
| 5 | cantnf.c | . . . . . . . . 9 ⊢ (𝜑 → 𝐶 ∈ (𝐴 ↑o 𝐵)) | |
| 6 | onelon 6409 | . . . . . . . . 9 ⊢ (((𝐴 ↑o 𝐵) ∈ On ∧ 𝐶 ∈ (𝐴 ↑o 𝐵)) → 𝐶 ∈ On) | |
| 7 | 4, 5, 6 | syl2anc 584 | . . . . . . . 8 ⊢ (𝜑 → 𝐶 ∈ On) |
| 8 | cantnf.e | . . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐶) | |
| 9 | ondif1 8539 | . . . . . . . 8 ⊢ (𝐶 ∈ (On ∖ 1o) ↔ (𝐶 ∈ On ∧ ∅ ∈ 𝐶)) | |
| 10 | 7, 8, 9 | sylanbrc 583 | . . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ (On ∖ 1o)) |
| 11 | 10 | eldifbd 3964 | . . . . . 6 ⊢ (𝜑 → ¬ 𝐶 ∈ 1o) |
| 12 | ssel 3977 | . . . . . . 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 198 | . . . . 5 ⊢ (𝜑 → ¬ (𝐴 ↑o 𝐵) ⊆ 1o) |
| 15 | oe0m 8556 | . . . . . . . . 9 ⊢ (𝐵 ∈ On → (∅ ↑o 𝐵) = (1o ∖ 𝐵)) | |
| 16 | 2, 15 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → (∅ ↑o 𝐵) = (1o ∖ 𝐵)) |
| 17 | difss 4136 | . . . . . . . 8 ⊢ (1o ∖ 𝐵) ⊆ 1o | |
| 18 | 16, 17 | eqsstrdi 4028 | . . . . . . 7 ⊢ (𝜑 → (∅ ↑o 𝐵) ⊆ 1o) |
| 19 | oveq1 7438 | . . . . . . . 8 ⊢ (𝐴 = ∅ → (𝐴 ↑o 𝐵) = (∅ ↑o 𝐵)) | |
| 20 | 19 | sseq1d 4015 | . . . . . . 7 ⊢ (𝐴 = ∅ → ((𝐴 ↑o 𝐵) ⊆ 1o ↔ (∅ ↑o 𝐵) ⊆ 1o)) |
| 21 | 18, 20 | syl5ibrcom 247 | . . . . . 6 ⊢ (𝜑 → (𝐴 = ∅ → (𝐴 ↑o 𝐵) ⊆ 1o)) |
| 22 | oe1m 8583 | . . . . . . . 8 ⊢ (𝐵 ∈ On → (1o ↑o 𝐵) = 1o) | |
| 23 | eqimss 4042 | . . . . . . . 8 ⊢ ((1o ↑o 𝐵) = 1o → (1o ↑o 𝐵) ⊆ 1o) | |
| 24 | 2, 22, 23 | 3syl 18 | . . . . . . 7 ⊢ (𝜑 → (1o ↑o 𝐵) ⊆ 1o) |
| 25 | oveq1 7438 | . . . . . . . 8 ⊢ (𝐴 = 1o → (𝐴 ↑o 𝐵) = (1o ↑o 𝐵)) | |
| 26 | 25 | sseq1d 4015 | . . . . . . 7 ⊢ (𝐴 = 1o → ((𝐴 ↑o 𝐵) ⊆ 1o ↔ (1o ↑o 𝐵) ⊆ 1o)) |
| 27 | 24, 26 | syl5ibrcom 247 | . . . . . 6 ⊢ (𝜑 → (𝐴 = 1o → (𝐴 ↑o 𝐵) ⊆ 1o)) |
| 28 | 21, 27 | jaod 860 | . . . . 5 ⊢ (𝜑 → ((𝐴 = ∅ ∨ 𝐴 = 1o) → (𝐴 ↑o 𝐵) ⊆ 1o)) |
| 29 | 14, 28 | mtod 198 | . . . 4 ⊢ (𝜑 → ¬ (𝐴 = ∅ ∨ 𝐴 = 1o)) |
| 30 | elpri 4649 | . . . . 5 ⊢ (𝐴 ∈ {∅, 1o} → (𝐴 = ∅ ∨ 𝐴 = 1o)) | |
| 31 | df2o3 8514 | . . . . 5 ⊢ 2o = {∅, 1o} | |
| 32 | 30, 31 | eleq2s 2859 | . . . 4 ⊢ (𝐴 ∈ 2o → (𝐴 = ∅ ∨ 𝐴 = 1o)) |
| 33 | 29, 32 | nsyl 140 | . . 3 ⊢ (𝜑 → ¬ 𝐴 ∈ 2o) |
| 34 | 1, 33 | eldifd 3962 | . 2 ⊢ (𝜑 → 𝐴 ∈ (On ∖ 2o)) |
| 35 | 34, 10 | jca 511 | 1 ⊢ (𝜑 → (𝐴 ∈ (On ∖ 2o) ∧ 𝐶 ∈ (On ∖ 1o))) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∨ wo 848 = wceq 1540 ∈ wcel 2108 ∀wral 3061 ∃wrex 3070 ∖ cdif 3948 ⊆ wss 3951 ∅c0 4333 {cpr 4628 {copab 5205 dom cdm 5685 ran crn 5686 Oncon0 6384 ‘cfv 6561 (class class class)co 7431 1oc1o 8499 2oc2o 8500 ↑o coe 8505 CNF ccnf 9701 |
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-rep 5279 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-csb 3900 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-mpt 5226 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-2nd 8015 df-frecs 8306 df-wrecs 8337 df-recs 8411 df-rdg 8450 df-1o 8506 df-2o 8507 df-oadd 8510 df-omul 8511 df-oexp 8512 |
| This theorem is referenced by: cantnflem3 9731 cantnflem4 9732 |
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