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Mirrors > Home > MPE Home > Th. List > cplem1 | Structured version Visualization version GIF version |
Description: Lemma for the Collection Principle cp 9308. (Contributed by NM, 17-Oct-2003.) |
Ref | Expression |
---|---|
cplem1.1 | ⊢ 𝐶 = {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)} |
cplem1.2 | ⊢ 𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶 |
Ref | Expression |
---|---|
cplem1 | ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝐷) ≠ ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | scott0 9303 | . . . . . 6 ⊢ (𝐵 = ∅ ↔ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅) | |
2 | cplem1.1 | . . . . . . 7 ⊢ 𝐶 = {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)} | |
3 | 2 | eqeq1i 2823 | . . . . . 6 ⊢ (𝐶 = ∅ ↔ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅) |
4 | 1, 3 | bitr4i 279 | . . . . 5 ⊢ (𝐵 = ∅ ↔ 𝐶 = ∅) |
5 | 4 | necon3bii 3065 | . . . 4 ⊢ (𝐵 ≠ ∅ ↔ 𝐶 ≠ ∅) |
6 | n0 4307 | . . . 4 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝐶) | |
7 | 5, 6 | bitri 276 | . . 3 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝐶) |
8 | 2 | ssrab3 4054 | . . . . . . . 8 ⊢ 𝐶 ⊆ 𝐵 |
9 | 8 | sseli 3960 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → 𝑤 ∈ 𝐵) |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → 𝑤 ∈ 𝐵)) |
11 | ssiun2 4962 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶) | |
12 | cplem1.2 | . . . . . . . 8 ⊢ 𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶 | |
13 | 11, 12 | sseqtrrdi 4015 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ⊆ 𝐷) |
14 | 13 | sseld 3963 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → 𝑤 ∈ 𝐷)) |
15 | 10, 14 | jcad 513 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → (𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷))) |
16 | inelcm 4410 | . . . . 5 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) → (𝐵 ∩ 𝐷) ≠ ∅) | |
17 | 15, 16 | syl6 35 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → (𝐵 ∩ 𝐷) ≠ ∅)) |
18 | 17 | exlimdv 1925 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∃𝑤 𝑤 ∈ 𝐶 → (𝐵 ∩ 𝐷) ≠ ∅)) |
19 | 7, 18 | syl5bi 243 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ≠ ∅ → (𝐵 ∩ 𝐷) ≠ ∅)) |
20 | 19 | rgen 3145 | 1 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝐷) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 = wceq 1528 ∃wex 1771 ∈ wcel 2105 ≠ wne 3013 ∀wral 3135 {crab 3139 ∩ cin 3932 ⊆ wss 3933 ∅c0 4288 ∪ ciun 4910 ‘cfv 6348 rankcrnk 9180 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-iin 4913 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-om 7570 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-r1 9181 df-rank 9182 |
This theorem is referenced by: cplem2 9307 |
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