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| Mirrors > Home > MPE Home > Th. List > fin1a2lem5 | Structured version Visualization version GIF version | ||
| Description: Lemma for fin1a2 10455. (Contributed by Stefan O'Rear, 7-Nov-2014.) |
| Ref | Expression |
|---|---|
| fin1a2lem.b | ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥)) |
| Ref | Expression |
|---|---|
| fin1a2lem5 | ⊢ (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸)) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | nneob 8694 | . 2 ⊢ (𝐴 ∈ ω → (∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎) ↔ ¬ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎))) | |
| 2 | fin1a2lem.b | . . . . . 6 ⊢ 𝐸 = (𝑥 ∈ ω ↦ (2o ·o 𝑥)) | |
| 3 | 2 | fin1a2lem4 10443 | . . . . 5 ⊢ 𝐸:ω–1-1→ω |
| 4 | f1fn 6805 | . . . . 5 ⊢ (𝐸:ω–1-1→ω → 𝐸 Fn ω) | |
| 5 | 3, 4 | ax-mp 5 | . . . 4 ⊢ 𝐸 Fn ω |
| 6 | fvelrnb 6969 | . . . 4 ⊢ (𝐸 Fn ω → (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω (𝐸‘𝑎) = 𝐴)) | |
| 7 | 5, 6 | ax-mp 5 | . . 3 ⊢ (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω (𝐸‘𝑎) = 𝐴) |
| 8 | eqcom 2744 | . . . . 5 ⊢ ((𝐸‘𝑎) = 𝐴 ↔ 𝐴 = (𝐸‘𝑎)) | |
| 9 | 2 | fin1a2lem3 10442 | . . . . . 6 ⊢ (𝑎 ∈ ω → (𝐸‘𝑎) = (2o ·o 𝑎)) |
| 10 | 9 | eqeq2d 2748 | . . . . 5 ⊢ (𝑎 ∈ ω → (𝐴 = (𝐸‘𝑎) ↔ 𝐴 = (2o ·o 𝑎))) |
| 11 | 8, 10 | bitrid 283 | . . . 4 ⊢ (𝑎 ∈ ω → ((𝐸‘𝑎) = 𝐴 ↔ 𝐴 = (2o ·o 𝑎))) |
| 12 | 11 | rexbiia 3092 | . . 3 ⊢ (∃𝑎 ∈ ω (𝐸‘𝑎) = 𝐴 ↔ ∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎)) |
| 13 | 7, 12 | bitri 275 | . 2 ⊢ (𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω 𝐴 = (2o ·o 𝑎)) |
| 14 | fvelrnb 6969 | . . . . 5 ⊢ (𝐸 Fn ω → (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω (𝐸‘𝑎) = suc 𝐴)) | |
| 15 | 5, 14 | ax-mp 5 | . . . 4 ⊢ (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω (𝐸‘𝑎) = suc 𝐴) |
| 16 | eqcom 2744 | . . . . . 6 ⊢ ((𝐸‘𝑎) = suc 𝐴 ↔ suc 𝐴 = (𝐸‘𝑎)) | |
| 17 | 9 | eqeq2d 2748 | . . . . . 6 ⊢ (𝑎 ∈ ω → (suc 𝐴 = (𝐸‘𝑎) ↔ suc 𝐴 = (2o ·o 𝑎))) |
| 18 | 16, 17 | bitrid 283 | . . . . 5 ⊢ (𝑎 ∈ ω → ((𝐸‘𝑎) = suc 𝐴 ↔ suc 𝐴 = (2o ·o 𝑎))) |
| 19 | 18 | rexbiia 3092 | . . . 4 ⊢ (∃𝑎 ∈ ω (𝐸‘𝑎) = suc 𝐴 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎)) |
| 20 | 15, 19 | bitri 275 | . . 3 ⊢ (suc 𝐴 ∈ ran 𝐸 ↔ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎)) |
| 21 | 20 | notbii 320 | . 2 ⊢ (¬ suc 𝐴 ∈ ran 𝐸 ↔ ¬ ∃𝑎 ∈ ω suc 𝐴 = (2o ·o 𝑎)) |
| 22 | 1, 13, 21 | 3bitr4g 314 | 1 ⊢ (𝐴 ∈ ω → (𝐴 ∈ ran 𝐸 ↔ ¬ suc 𝐴 ∈ ran 𝐸)) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 = wceq 1540 ∈ wcel 2108 ∃wrex 3070 ↦ cmpt 5225 ran crn 5686 suc csuc 6386 Fn wfn 6556 –1-1→wf1 6558 ‘cfv 6561 (class class class)co 7431 ωcom 7887 2oc2o 8500 ·o comu 8504 |
| 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 |
| This theorem is referenced by: fin1a2lem6 10445 |
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