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Mirrors > Home > MPE Home > Th. List > cplem1 | Structured version Visualization version GIF version |
Description: Lemma for the Collection Principle cp 8637. (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 8632 | . . . . . 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 2615 | . . . . . 6 ⊢ (𝐶 = ∅ ↔ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅) |
4 | 1, 3 | bitr4i 266 | . . . . 5 ⊢ (𝐵 = ∅ ↔ 𝐶 = ∅) |
5 | 4 | necon3bii 2834 | . . . 4 ⊢ (𝐵 ≠ ∅ ↔ 𝐶 ≠ ∅) |
6 | n0 3890 | . . . 4 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝐶) | |
7 | 5, 6 | bitri 263 | . . 3 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝐶) |
8 | ssrab2 3650 | . . . . . . . . 9 ⊢ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)} ⊆ 𝐵 | |
9 | 2, 8 | eqsstri 3598 | . . . . . . . 8 ⊢ 𝐶 ⊆ 𝐵 |
10 | 9 | sseli 3564 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → 𝑤 ∈ 𝐵) |
11 | 10 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → 𝑤 ∈ 𝐵)) |
12 | ssiun2 4499 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶) | |
13 | cplem1.2 | . . . . . . . 8 ⊢ 𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶 | |
14 | 12, 13 | syl6sseqr 3615 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ⊆ 𝐷) |
15 | 14 | sseld 3567 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → 𝑤 ∈ 𝐷)) |
16 | 11, 15 | jcad 554 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → (𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷))) |
17 | inelcm 3984 | . . . . 5 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) → (𝐵 ∩ 𝐷) ≠ ∅) | |
18 | 16, 17 | syl6 34 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → (𝐵 ∩ 𝐷) ≠ ∅)) |
19 | 18 | exlimdv 1848 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∃𝑤 𝑤 ∈ 𝐶 → (𝐵 ∩ 𝐷) ≠ ∅)) |
20 | 7, 19 | syl5bi 231 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ≠ ∅ → (𝐵 ∩ 𝐷) ≠ ∅)) |
21 | 20 | rgen 2906 | 1 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝐷) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 {crab 2900 ∩ cin 3539 ⊆ wss 3540 ∅c0 3874 ∪ ciun 4455 ‘cfv 5804 rankcrnk 8509 |
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-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-int 4411 df-iun 4457 df-iin 4458 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-om 6958 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-r1 8510 df-rank 8511 |
This theorem is referenced by: cplem2 8636 |
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