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Mirrors > Home > MPE Home > Th. List > infeq5i | Structured version Visualization version GIF version |
Description: Half of infeq5 8701. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
infeq5i | ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difexg 4943 | . 2 ⊢ (ω ∈ V → (ω ∖ {∅}) ∈ V) | |
2 | 0ex 4925 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | snid 4348 | . . . 4 ⊢ ∅ ∈ {∅} |
4 | disj4 4170 | . . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊊ ω) | |
5 | disj3 4165 | . . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅})) | |
6 | 4, 5 | bitr3i 266 | . . . . 5 ⊢ (¬ (ω ∖ {∅}) ⊊ ω ↔ ω = (ω ∖ {∅})) |
7 | peano1 7235 | . . . . . . 7 ⊢ ∅ ∈ ω | |
8 | eleq2 2839 | . . . . . . 7 ⊢ (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅}))) | |
9 | 7, 8 | mpbii 223 | . . . . . 6 ⊢ (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅})) |
10 | 9 | eldifbd 3736 | . . . . 5 ⊢ (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅}) |
11 | 6, 10 | sylbi 207 | . . . 4 ⊢ (¬ (ω ∖ {∅}) ⊊ ω → ¬ ∅ ∈ {∅}) |
12 | 3, 11 | mt4 116 | . . 3 ⊢ (ω ∖ {∅}) ⊊ ω |
13 | unidif0 4970 | . . . . 5 ⊢ ∪ (ω ∖ {∅}) = ∪ ω | |
14 | limom 7230 | . . . . . 6 ⊢ Lim ω | |
15 | limuni 5927 | . . . . . 6 ⊢ (Lim ω → ω = ∪ ω) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ ω = ∪ ω |
17 | 13, 16 | eqtr4i 2796 | . . . 4 ⊢ ∪ (ω ∖ {∅}) = ω |
18 | 17 | psseq2i 3847 | . . 3 ⊢ ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊊ ω) |
19 | 12, 18 | mpbir 221 | . 2 ⊢ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) |
20 | psseq1 3844 | . . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ 𝑥)) | |
21 | unieq 4583 | . . . . 5 ⊢ (𝑥 = (ω ∖ {∅}) → ∪ 𝑥 = ∪ (ω ∖ {∅})) | |
22 | 21 | psseq2d 3850 | . . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}))) |
23 | 20, 22 | bitrd 268 | . . 3 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}))) |
24 | 23 | spcegv 3445 | . 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 1631 ∃wex 1852 ∈ wcel 2145 Vcvv 3351 ∖ cdif 3720 ∩ cin 3722 ⊊ wpss 3724 ∅c0 4063 {csn 4317 ∪ cuni 4575 Lim wlim 5866 ωcom 7215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-sep 4916 ax-nul 4924 ax-pr 5035 ax-un 7099 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 837 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-rab 3070 df-v 3353 df-sbc 3588 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4227 df-pw 4300 df-sn 4318 df-pr 4320 df-tp 4322 df-op 4324 df-uni 4576 df-br 4788 df-opab 4848 df-tr 4888 df-eprel 5163 df-po 5171 df-so 5172 df-fr 5209 df-we 5211 df-ord 5868 df-on 5869 df-lim 5870 df-suc 5871 df-om 7216 |
This theorem is referenced by: infeq5 8701 inf5 8709 |
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