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Mirrors > Home > MPE Home > Th. List > infeq5i | Structured version Visualization version GIF version |
Description: Half of infeq5 8534. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
infeq5i | ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difexg 4808 | . 2 ⊢ (ω ∈ V → (ω ∖ {∅}) ∈ V) | |
2 | 0ex 4790 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | snid 4208 | . . . 4 ⊢ ∅ ∈ {∅} |
4 | disj4 4025 | . . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊊ ω) | |
5 | disj3 4021 | . . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅})) | |
6 | 4, 5 | bitr3i 266 | . . . . 5 ⊢ (¬ (ω ∖ {∅}) ⊊ ω ↔ ω = (ω ∖ {∅})) |
7 | peano1 7085 | . . . . . . 7 ⊢ ∅ ∈ ω | |
8 | eleq2 2690 | . . . . . . 7 ⊢ (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅}))) | |
9 | 7, 8 | mpbii 223 | . . . . . 6 ⊢ (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅})) |
10 | 9 | eldifbd 3587 | . . . . 5 ⊢ (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅}) |
11 | 6, 10 | sylbi 207 | . . . 4 ⊢ (¬ (ω ∖ {∅}) ⊊ ω → ¬ ∅ ∈ {∅}) |
12 | 3, 11 | mt4 115 | . . 3 ⊢ (ω ∖ {∅}) ⊊ ω |
13 | unidif0 4838 | . . . . 5 ⊢ ∪ (ω ∖ {∅}) = ∪ ω | |
14 | limom 7080 | . . . . . 6 ⊢ Lim ω | |
15 | limuni 5785 | . . . . . 6 ⊢ (Lim ω → ω = ∪ ω) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ ω = ∪ ω |
17 | 13, 16 | eqtr4i 2647 | . . . 4 ⊢ ∪ (ω ∖ {∅}) = ω |
18 | 17 | psseq2i 3697 | . . 3 ⊢ ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊊ ω) |
19 | 12, 18 | mpbir 221 | . 2 ⊢ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) |
20 | psseq1 3694 | . . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ 𝑥)) | |
21 | unieq 4444 | . . . . 5 ⊢ (𝑥 = (ω ∖ {∅}) → ∪ 𝑥 = ∪ (ω ∖ {∅})) | |
22 | 21 | psseq2d 3700 | . . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}))) |
23 | 20, 22 | bitrd 268 | . . 3 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}))) |
24 | 23 | spcegv 3294 | . 2 ⊢ ((ω ∖ {∅}) ∈ V → ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) → ∃𝑥 𝑥 ⊊ ∪ 𝑥)) |
25 | 1, 19, 24 | mpisyl 21 | 1 ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 = wceq 1483 ∃wex 1704 ∈ wcel 1990 Vcvv 3200 ∖ cdif 3571 ∩ cin 3573 ⊊ wpss 3575 ∅c0 3915 {csn 4177 ∪ cuni 4436 Lim wlim 5724 ωcom 7065 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-pss 3590 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-tp 4182 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-tr 4753 df-eprel 5029 df-po 5035 df-so 5036 df-fr 5073 df-we 5075 df-ord 5726 df-on 5727 df-lim 5728 df-suc 5729 df-om 7066 |
This theorem is referenced by: infeq5 8534 inf5 8542 |
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