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Mirrors > Home > MPE Home > Th. List > oaabslem | Structured version Visualization version GIF version |
Description: Lemma for oaabs 8635. (Contributed by NM, 9-Dec-2004.) |
Ref | Expression |
---|---|
oaabslem | ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 7848 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | limom 7858 | . . . . . 6 ⊢ Lim ω | |
3 | 2 | jctr 526 | . . . . 5 ⊢ (ω ∈ On → (ω ∈ On ∧ Lim ω)) |
4 | oalim 8519 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 +o ω) = ∪ 𝑥 ∈ ω (𝐴 +o 𝑥)) | |
5 | 1, 3, 4 | syl2an 597 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 +o ω) = ∪ 𝑥 ∈ ω (𝐴 +o 𝑥)) |
6 | ordom 7852 | . . . . . . . 8 ⊢ Ord ω | |
7 | nnacl 8599 | . . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈ ω) | |
8 | ordelss 6372 | . . . . . . . 8 ⊢ ((Ord ω ∧ (𝐴 +o 𝑥) ∈ ω) → (𝐴 +o 𝑥) ⊆ ω) | |
9 | 6, 7, 8 | sylancr 588 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ⊆ ω) |
10 | 9 | ralrimiva 3147 | . . . . . 6 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω) |
11 | iunss 5044 | . . . . . 6 ⊢ (∪ 𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω ↔ ∀𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω) | |
12 | 10, 11 | sylibr 233 | . . . . 5 ⊢ (𝐴 ∈ ω → ∪ 𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω) |
13 | 12 | adantr 482 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → ∪ 𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω) |
14 | 5, 13 | eqsstrd 4018 | . . 3 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 +o ω) ⊆ ω) |
15 | 14 | ancoms 460 | . 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) ⊆ ω) |
16 | oaword2 8541 | . . 3 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On) → ω ⊆ (𝐴 +o ω)) | |
17 | 1, 16 | sylan2 594 | . 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → ω ⊆ (𝐴 +o ω)) |
18 | 15, 17 | eqssd 3997 | 1 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3062 ⊆ wss 3946 ∪ ciun 4993 Ord word 6355 Oncon0 6356 Lim wlim 6357 (class class class)co 7396 ωcom 7842 +o coa 8450 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5281 ax-sep 5295 ax-nul 5302 ax-pr 5423 ax-un 7712 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3776 df-csb 3892 df-dif 3949 df-un 3951 df-in 3953 df-ss 3963 df-pss 3965 df-nul 4321 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4905 df-iun 4995 df-br 5145 df-opab 5207 df-mpt 5228 df-tr 5262 df-id 5570 df-eprel 5576 df-po 5584 df-so 5585 df-fr 5627 df-we 5629 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-pred 6292 df-ord 6359 df-on 6360 df-lim 6361 df-suc 6362 df-iota 6487 df-fun 6537 df-fn 6538 df-f 6539 df-f1 6540 df-fo 6541 df-f1o 6542 df-fv 6543 df-ov 7399 df-oprab 7400 df-mpo 7401 df-om 7843 df-2nd 7963 df-frecs 8253 df-wrecs 8284 df-recs 8358 df-rdg 8397 df-oadd 8457 |
This theorem is referenced by: oaabs 8635 oaabs2 8636 oancom 9633 1oaomeqom 41914 |
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