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Mirrors > Home > MPE Home > Th. List > oaabslem | Structured version Visualization version GIF version |
Description: Lemma for oaabs 8314. (Contributed by NM, 9-Dec-2004.) |
Ref | Expression |
---|---|
oaabslem | ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 7617 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | limom 7626 | . . . . . 6 ⊢ Lim ω | |
3 | 2 | jctr 528 | . . . . 5 ⊢ (ω ∈ On → (ω ∈ On ∧ Lim ω)) |
4 | oalim 8200 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 +o ω) = ∪ 𝑥 ∈ ω (𝐴 +o 𝑥)) | |
5 | 1, 3, 4 | syl2an 599 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 +o ω) = ∪ 𝑥 ∈ ω (𝐴 +o 𝑥)) |
6 | ordom 7620 | . . . . . . . 8 ⊢ Ord ω | |
7 | nnacl 8280 | . . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈ ω) | |
8 | ordelss 6198 | . . . . . . . 8 ⊢ ((Ord ω ∧ (𝐴 +o 𝑥) ∈ ω) → (𝐴 +o 𝑥) ⊆ ω) | |
9 | 6, 7, 8 | sylancr 590 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ⊆ ω) |
10 | 9 | ralrimiva 3097 | . . . . . 6 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω) |
11 | iunss 4941 | . . . . . 6 ⊢ (∪ 𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω ↔ ∀𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω) | |
12 | 10, 11 | sylibr 237 | . . . . 5 ⊢ (𝐴 ∈ ω → ∪ 𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω) |
13 | 12 | adantr 484 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → ∪ 𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω) |
14 | 5, 13 | eqsstrd 3925 | . . 3 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 +o ω) ⊆ ω) |
15 | 14 | ancoms 462 | . 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) ⊆ ω) |
16 | oaword2 8222 | . . 3 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On) → ω ⊆ (𝐴 +o ω)) | |
17 | 1, 16 | sylan2 596 | . 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → ω ⊆ (𝐴 +o ω)) |
18 | 15, 17 | eqssd 3904 | 1 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1542 ∈ wcel 2114 ∀wral 3054 ⊆ wss 3853 ∪ ciun 4891 Ord word 6181 Oncon0 6182 Lim wlim 6183 (class class class)co 7182 ωcom 7611 +o coa 8140 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5306 ax-un 7491 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4222 df-if 4425 df-pw 4500 df-sn 4527 df-pr 4529 df-tp 4531 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-tr 5147 df-id 5439 df-eprel 5444 df-po 5452 df-so 5453 df-fr 5493 df-we 5495 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-pred 6139 df-ord 6185 df-on 6186 df-lim 6187 df-suc 6188 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7185 df-oprab 7186 df-mpo 7187 df-om 7612 df-wrecs 7988 df-recs 8049 df-rdg 8087 df-oadd 8147 |
This theorem is referenced by: oaabs 8314 oaabs2 8315 oancom 9199 |
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