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Mirrors > Home > MPE Home > Th. List > infeq5i | Structured version Visualization version GIF version |
Description: Half of infeq5 9088. (Contributed by Mario Carneiro, 16-Nov-2014.) |
Ref | Expression |
---|---|
infeq5i | ⊢ (ω ∈ V → ∃𝑥 𝑥 ⊊ ∪ 𝑥) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | difexg 5222 | . 2 ⊢ (ω ∈ V → (ω ∖ {∅}) ∈ V) | |
2 | 0ex 5202 | . . . . 5 ⊢ ∅ ∈ V | |
3 | 2 | snid 4591 | . . . 4 ⊢ ∅ ∈ {∅} |
4 | disj4 4404 | . . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ¬ (ω ∖ {∅}) ⊊ ω) | |
5 | disj3 4399 | . . . . . 6 ⊢ ((ω ∩ {∅}) = ∅ ↔ ω = (ω ∖ {∅})) | |
6 | 4, 5 | bitr3i 278 | . . . . 5 ⊢ (¬ (ω ∖ {∅}) ⊊ ω ↔ ω = (ω ∖ {∅})) |
7 | peano1 7590 | . . . . . . 7 ⊢ ∅ ∈ ω | |
8 | eleq2 2898 | . . . . . . 7 ⊢ (ω = (ω ∖ {∅}) → (∅ ∈ ω ↔ ∅ ∈ (ω ∖ {∅}))) | |
9 | 7, 8 | mpbii 234 | . . . . . 6 ⊢ (ω = (ω ∖ {∅}) → ∅ ∈ (ω ∖ {∅})) |
10 | 9 | eldifbd 3946 | . . . . 5 ⊢ (ω = (ω ∖ {∅}) → ¬ ∅ ∈ {∅}) |
11 | 6, 10 | sylbi 218 | . . . 4 ⊢ (¬ (ω ∖ {∅}) ⊊ ω → ¬ ∅ ∈ {∅}) |
12 | 3, 11 | mt4 116 | . . 3 ⊢ (ω ∖ {∅}) ⊊ ω |
13 | unidif0 5251 | . . . . 5 ⊢ ∪ (ω ∖ {∅}) = ∪ ω | |
14 | limom 7584 | . . . . . 6 ⊢ Lim ω | |
15 | limuni 6244 | . . . . . 6 ⊢ (Lim ω → ω = ∪ ω) | |
16 | 14, 15 | ax-mp 5 | . . . . 5 ⊢ ω = ∪ ω |
17 | 13, 16 | eqtr4i 2844 | . . . 4 ⊢ ∪ (ω ∖ {∅}) = ω |
18 | 17 | psseq2i 4064 | . . 3 ⊢ ((ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) ↔ (ω ∖ {∅}) ⊊ ω) |
19 | 12, 18 | mpbir 232 | . 2 ⊢ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}) |
20 | psseq1 4061 | . . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ 𝑥)) | |
21 | unieq 4838 | . . . . 5 ⊢ (𝑥 = (ω ∖ {∅}) → ∪ 𝑥 = ∪ (ω ∖ {∅})) | |
22 | 21 | psseq2d 4067 | . . . 4 ⊢ (𝑥 = (ω ∖ {∅}) → ((ω ∖ {∅}) ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}))) |
23 | 20, 22 | bitrd 280 | . . 3 ⊢ (𝑥 = (ω ∖ {∅}) → (𝑥 ⊊ ∪ 𝑥 ↔ (ω ∖ {∅}) ⊊ ∪ (ω ∖ {∅}))) |
24 | 23 | spcegv 3594 | . 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 1528 ∃wex 1771 ∈ wcel 2105 Vcvv 3492 ∖ cdif 3930 ∩ cin 3932 ⊊ wpss 3934 ∅c0 4288 {csn 4557 ∪ cuni 4830 Lim wlim 6185 ωcom 7569 |
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-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-rab 3144 df-v 3494 df-sbc 3770 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-br 5058 df-opab 5120 df-tr 5164 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-om 7570 |
This theorem is referenced by: infeq5 9088 inf5 9096 |
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