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Mirrors > Home > MPE Home > Th. List > oaabslem | Structured version Visualization version GIF version |
Description: Lemma for oaabs 7611. (Contributed by NM, 9-Dec-2004.) |
Ref | Expression |
---|---|
oaabslem | ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 ω) = ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 6963 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | limom 6972 | . . . . . 6 ⊢ Lim ω | |
3 | 2 | jctr 563 | . . . . 5 ⊢ (ω ∈ On → (ω ∈ On ∧ Lim ω)) |
4 | oalim 7499 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 +𝑜 ω) = ∪ 𝑥 ∈ ω (𝐴 +𝑜 𝑥)) | |
5 | 1, 3, 4 | syl2an 493 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 +𝑜 ω) = ∪ 𝑥 ∈ ω (𝐴 +𝑜 𝑥)) |
6 | ordom 6966 | . . . . . . . 8 ⊢ Ord ω | |
7 | nnacl 7578 | . . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 𝑥) ∈ ω) | |
8 | ordelss 5656 | . . . . . . . 8 ⊢ ((Ord ω ∧ (𝐴 +𝑜 𝑥) ∈ ω) → (𝐴 +𝑜 𝑥) ⊆ ω) | |
9 | 6, 7, 8 | sylancr 694 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +𝑜 𝑥) ⊆ ω) |
10 | 9 | ralrimiva 2949 | . . . . . 6 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ ω (𝐴 +𝑜 𝑥) ⊆ ω) |
11 | iunss 4497 | . . . . . 6 ⊢ (∪ 𝑥 ∈ ω (𝐴 +𝑜 𝑥) ⊆ ω ↔ ∀𝑥 ∈ ω (𝐴 +𝑜 𝑥) ⊆ ω) | |
12 | 10, 11 | sylibr 223 | . . . . 5 ⊢ (𝐴 ∈ ω → ∪ 𝑥 ∈ ω (𝐴 +𝑜 𝑥) ⊆ ω) |
13 | 12 | adantr 480 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → ∪ 𝑥 ∈ ω (𝐴 +𝑜 𝑥) ⊆ ω) |
14 | 5, 13 | eqsstrd 3602 | . . 3 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 +𝑜 ω) ⊆ ω) |
15 | 14 | ancoms 468 | . 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 ω) ⊆ ω) |
16 | oaword2 7520 | . . 3 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On) → ω ⊆ (𝐴 +𝑜 ω)) | |
17 | 1, 16 | sylan2 490 | . 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → ω ⊆ (𝐴 +𝑜 ω)) |
18 | 15, 17 | eqssd 3585 | 1 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +𝑜 ω) = ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ∀wral 2896 ⊆ wss 3540 ∪ ciun 4455 Ord word 5639 Oncon0 5640 Lim wlim 5641 (class class class)co 6549 ωcom 6957 +𝑜 coa 7444 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-oadd 7451 |
This theorem is referenced by: oaabs 7611 oaabs2 7612 oancom 8431 |
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