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Mirrors > Home > MPE Home > Th. List > cplem1 | Structured version Visualization version GIF version |
Description: Lemma for the Collection Principle cp 9472. (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 9467 | . . . . . 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 2741 | . . . . . 6 ⊢ (𝐶 = ∅ ↔ {𝑦 ∈ 𝐵 ∣ ∀𝑧 ∈ 𝐵 (rank‘𝑦) ⊆ (rank‘𝑧)} = ∅) |
4 | 1, 3 | bitr4i 281 | . . . . 5 ⊢ (𝐵 = ∅ ↔ 𝐶 = ∅) |
5 | 4 | necon3bii 2984 | . . . 4 ⊢ (𝐵 ≠ ∅ ↔ 𝐶 ≠ ∅) |
6 | n0 4247 | . . . 4 ⊢ (𝐶 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝐶) | |
7 | 5, 6 | bitri 278 | . . 3 ⊢ (𝐵 ≠ ∅ ↔ ∃𝑤 𝑤 ∈ 𝐶) |
8 | 2 | ssrab3 3981 | . . . . . . . 8 ⊢ 𝐶 ⊆ 𝐵 |
9 | 8 | sseli 3883 | . . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → 𝑤 ∈ 𝐵) |
10 | 9 | a1i 11 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → 𝑤 ∈ 𝐵)) |
11 | ssiun2 4942 | . . . . . . . 8 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶) | |
12 | cplem1.2 | . . . . . . . 8 ⊢ 𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶 | |
13 | 11, 12 | sseqtrrdi 3938 | . . . . . . 7 ⊢ (𝑥 ∈ 𝐴 → 𝐶 ⊆ 𝐷) |
14 | 13 | sseld 3886 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → 𝑤 ∈ 𝐷)) |
15 | 10, 14 | jcad 516 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → (𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷))) |
16 | inelcm 4365 | . . . . 5 ⊢ ((𝑤 ∈ 𝐵 ∧ 𝑤 ∈ 𝐷) → (𝐵 ∩ 𝐷) ≠ ∅) | |
17 | 15, 16 | syl6 35 | . . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑤 ∈ 𝐶 → (𝐵 ∩ 𝐷) ≠ ∅)) |
18 | 17 | exlimdv 1941 | . . 3 ⊢ (𝑥 ∈ 𝐴 → (∃𝑤 𝑤 ∈ 𝐶 → (𝐵 ∩ 𝐷) ≠ ∅)) |
19 | 7, 18 | syl5bi 245 | . 2 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ≠ ∅ → (𝐵 ∩ 𝐷) ≠ ∅)) |
20 | 19 | rgen 3061 | 1 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ≠ ∅ → (𝐵 ∩ 𝐷) ≠ ∅) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 = wceq 1543 ∃wex 1787 ∈ wcel 2112 ≠ wne 2932 ∀wral 3051 {crab 3055 ∩ cin 3852 ⊆ wss 3853 ∅c0 4223 ∪ ciun 4890 ‘cfv 6358 rankcrnk 9344 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-int 4846 df-iun 4892 df-iin 4893 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-om 7623 df-wrecs 8025 df-recs 8086 df-rdg 8124 df-r1 9345 df-rank 9346 |
This theorem is referenced by: cplem2 9471 |
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