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Mirrors > Home > MPE Home > Th. List > oaabslem | Structured version Visualization version GIF version |
Description: Lemma for oaabs 7991. (Contributed by NM, 9-Dec-2004.) |
Ref | Expression |
---|---|
oaabslem | ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 7332 | . . . . 5 ⊢ (𝐴 ∈ ω → 𝐴 ∈ On) | |
2 | limom 7341 | . . . . . 6 ⊢ Lim ω | |
3 | 2 | jctr 522 | . . . . 5 ⊢ (ω ∈ On → (ω ∈ On ∧ Lim ω)) |
4 | oalim 7879 | . . . . 5 ⊢ ((𝐴 ∈ On ∧ (ω ∈ On ∧ Lim ω)) → (𝐴 +o ω) = ∪ 𝑥 ∈ ω (𝐴 +o 𝑥)) | |
5 | 1, 3, 4 | syl2an 591 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 +o ω) = ∪ 𝑥 ∈ ω (𝐴 +o 𝑥)) |
6 | ordom 7335 | . . . . . . . 8 ⊢ Ord ω | |
7 | nnacl 7958 | . . . . . . . 8 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ∈ ω) | |
8 | ordelss 5979 | . . . . . . . 8 ⊢ ((Ord ω ∧ (𝐴 +o 𝑥) ∈ ω) → (𝐴 +o 𝑥) ⊆ ω) | |
9 | 6, 7, 8 | sylancr 583 | . . . . . . 7 ⊢ ((𝐴 ∈ ω ∧ 𝑥 ∈ ω) → (𝐴 +o 𝑥) ⊆ ω) |
10 | 9 | ralrimiva 3175 | . . . . . 6 ⊢ (𝐴 ∈ ω → ∀𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω) |
11 | iunss 4781 | . . . . . 6 ⊢ (∪ 𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω ↔ ∀𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω) | |
12 | 10, 11 | sylibr 226 | . . . . 5 ⊢ (𝐴 ∈ ω → ∪ 𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω) |
13 | 12 | adantr 474 | . . . 4 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → ∪ 𝑥 ∈ ω (𝐴 +o 𝑥) ⊆ ω) |
14 | 5, 13 | eqsstrd 3864 | . . 3 ⊢ ((𝐴 ∈ ω ∧ ω ∈ On) → (𝐴 +o ω) ⊆ ω) |
15 | 14 | ancoms 452 | . 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) ⊆ ω) |
16 | oaword2 7900 | . . 3 ⊢ ((ω ∈ On ∧ 𝐴 ∈ On) → ω ⊆ (𝐴 +o ω)) | |
17 | 1, 16 | sylan2 588 | . 2 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → ω ⊆ (𝐴 +o ω)) |
18 | 15, 17 | eqssd 3844 | 1 ⊢ ((ω ∈ On ∧ 𝐴 ∈ ω) → (𝐴 +o ω) = ω) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∀wral 3117 ⊆ wss 3798 ∪ ciun 4740 Ord word 5962 Oncon0 5963 Lim wlim 5964 (class class class)co 6905 ωcom 7326 +o coa 7823 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-8 2168 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pow 5065 ax-pr 5127 ax-un 7209 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3or 1114 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-pss 3814 df-nul 4145 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-tr 4976 df-id 5250 df-eprel 5255 df-po 5263 df-so 5264 df-fr 5301 df-we 5303 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-pred 5920 df-ord 5966 df-on 5967 df-lim 5968 df-suc 5969 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 df-ov 6908 df-oprab 6909 df-mpt2 6910 df-om 7327 df-wrecs 7672 df-recs 7734 df-rdg 7772 df-oadd 7830 |
This theorem is referenced by: oaabs 7991 oaabs2 7992 oancom 8825 |
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